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probability | How to find densitiy of |X| | https://math.stackexchange.com/questions/4163938/how-to-find-densitiy-of-x | <p>Let <span class="math-container">$X \sim \operatorname{N}\left(0,1\right)$</span> be a standard normal distributed random variable:
Can someone please please show me the whole way how to find the density of <span class="math-container">$\left\vert X\right\vert$</span>.</p>
<p><em>My tries:</em> I saw sites where som... | <p>A standard normally distributed random variable has a symmetric distribution about <span class="math-container">$0$</span>, so <span class="math-container">$|X|$</span> has zero density below <span class="math-container">$0$</span>, and double the standard normal density above <span class="math-container">$0$</span>... | 300 |
probability | From $\Pr[A\cap B]$ and $\Pr[\bar{A}\cap B]$ deduce $\Pr[A]$ | https://math.stackexchange.com/questions/4165711/from-pra-cap-b-and-pr-bara-cap-b-deduce-pra | <p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be two events such that <span class="math-container">$\Pr[A\cap B]=0.2$</span> and <span class="math-container">$0.3<\Pr[\bar{A}\cap B]<0.4$</span>. Find <span class="math-container">$a,b\in\mathbb{R}$</span> such that <spa... | <p>first observe that</p>
<p><span class="math-container">$$(A\cap B)\cup(\overline{A}\cap B)=B$$</span></p>
<p>Thus</p>
<p><span class="math-container">$$0.5< \mathbb{P}[B]< 0.6$$</span></p>
<p>and evidently</p>
<p><span class="math-container">$$0.2\leq \mathbb{P}[A]\leq 0.7$$</span></p>
<p>here is a Venn's diag... | 301 |
probability | Finding a rare biased coin from an infinite set | https://math.stackexchange.com/questions/4160412/finding-a-rare-biased-coin-from-an-infinite-set | <p>I'm trying to develop an algorithm for finding biased coins. The basic problem formulation is this:</p>
<ol>
<li>There are an infinite number of coins</li>
<li>Some proportion <span class="math-container">$t$</span> of the coins is biased (this number is known)</li>
<li>All biased coins have the same probability <sp... | <blockquote>
<p>Is it possible, for instance, to abandon a coin before n flips is reached based on some criteria, i.e. using the evidence collected so far to judge whether it is worthwhile to keep flipping that coin or move on to another? It seems like the value of t, particularly if it is low, should be a useful prior... | 302 |
probability | A bag contains 6 white balls, 5 black balls and 2 red balls. | https://math.stackexchange.com/questions/4165726/a-bag-contains-6-white-balls-5-black-balls-and-2-red-balls | <p>A bag contains 6 white balls, 5 black balls and 2 red balls.
If two balls are drawn at random, what is the probability that neither of them are white?</p>
<p>For this question, the method that I used was to consider the four possible cases, BB, RR, BR, RB.</p>
<p>Therefore <span class="math-container">$$P = P(BB)+P(... | 303 | |
probability | What is probability of choose already choosen when we choose n values from set limited to m (n <= m)? | https://math.stackexchange.com/questions/4166183/what-is-probability-of-choose-already-choosen-when-we-choose-n-values-from-set-l | <p>I study math many years ago (I am engineer) and not remember how exactly calculate probability for such problem. I choose n values from m set and want calculate probability that all will be unique, has <span class="math-container">$1$</span> repeat or less, <span class="math-container">$2$</span> repeats or less or ... | <p>To calculate the probability of no repeats, observe</p>
<p><em>Total ordered sequences</em> = <span class="math-container">$m\times m\times \dots\times m = m^n $</span></p>
<p><em>Number of sequences with each element distinct</em> = <span class="math-container">$\binom{m}{n} \times n!=\frac{m!}{(m-n)!} $</span></p>... | 304 |
probability | $\sum E(|X_n - X|^r)$ converges implies almost sure convergence | https://math.stackexchange.com/questions/2770386/sum-ex-n-xr-converges-implies-almost-sure-convergence | <p>How do we show that if $\sum E(|X_n - X|^r) < \infty$ then $X_n {\to} X$ almost surely for $r > 0$?</p>
<p>I know that it's true for $\sum P(|X_n - X| > \epsilon) < \infty$, but how do we extend this to account for the $r$-th mean?</p>
| <p>$\sum E(|X_{n}-X|^{r}) = E(\sum |X_{n}-X|^{r})$. So $\sum|X_{n}-X|^{r} < \infty$ a.s $\implies$ that $|X_{n}-X|^{r} \rightarrow 0$ a.s and therefore that $X_{n} \rightarrow X$ a.s </p>
| 305 |
probability | Use of binomial probability distribution if events occur simultaneously? | https://math.stackexchange.com/questions/4166345/use-of-binomial-probability-distribution-if-events-occur-simultaneously | <p>I recently did a question, Numbers are selected at random, one at a time from two-digit numbers {00-99} with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, Find the probability that the event E occurs at least 3 times.</p>
<p>The ... | <p>The <em>count of successes</em> among a <em>specified amount</em> of <em>Bernoulli trials</em> each with an <em>independent and identically distributed</em> success rate, is a <strong>Binomially Distributed</strong> random variable.</p>
<p>Order of the sequence is not one of these criteria.</p>
<p><strong>However</s... | 306 |
probability | Coin tosses probability to calculate | https://math.stackexchange.com/questions/4107396/coin-tosses-probability-to-calculate | <p>I'm stuck at solving the following problem: launch 3 fair coins independently. Let A the event: "you get at least a head" and B "you get exactly one tail".
Then what is the probability of the event <span class="math-container">$A \cup B$</span>?</p>
| <p>Note that <span class="math-container">$\overline{A\cup B}=\overline{A}\cap \overline{B}$</span>. Now <span class="math-container">$\overline{A}$</span> is you get not heads and <span class="math-container">$\overline{B}$</span> is you get any number of tails but one. Only one option fits <span class="math-container... | 307 |
probability | Conditional Probability : what is the prob. that all are girls given that there is atleast one girl named Lila. | https://math.stackexchange.com/questions/1893041/conditional-probability-what-is-the-prob-that-all-are-girls-given-that-there | <p><strong>Q.1)</strong> A family has $n$ children, $n\geq2$. We ask from the father, "Do you have at least one daughter named Lilia?" He replies, "Yes!". What is the probability that all of their children are girls? </p>
<p>In other words, we want to find the probability that all $n$ children are girls, given that th... | <p>Alright so the first question seems to be confusing some people.
Let $B$ be the event of all boys. The complement of $B$ is $B^c$, which is the event of at least one girl. Here is where confusion then plays out: is $B^c$ the same as a girl named Lilia? Does this mean that if you have at least one girl, then her nam... | 308 |
probability | Probability of Heads in a coin | https://math.stackexchange.com/questions/144499/probability-of-heads-in-a-coin | <p>I was wondering, if you flip a fair coin $5$ times, whether you can calculate the probability of getting at least one head is calculated like this:</p>
<p>You can do the complement of getting at least one head which is TTTTT: $\dfrac1{2^5} =\dfrac1{32}$</p>
<p>Then you do $$1-\frac1{32}= \frac{31}{32}\;,$$ so that... | <p>Yes, that's correct. This technique is often called <em>complementary counting</em>.</p>
| 309 |
probability | How to calculate "at most" with special cases removed? | https://math.stackexchange.com/questions/2972892/how-to-calculate-at-most-with-special-cases-removed | <p>I am trying to calculate the percentage of winning for a certain event but cannot find the right approach or an easier way to exclude special cases. </p>
<p>Problem::
In many Trading Card Games (TCG) players are given the option to enter tournaments that reward them based on the number of wins they can achieve. Be... | <p>I'm not sure I understand the problem, but what I think you're saying is that you play a series of games, each of which you have a probability of 1/2 of winning, and the series ends whenever your cumulative score reaches -3 or +7, which ever comes first.</p>
<p>If this interpretation is correct, then another way to... | 310 |
probability | Conditional probability question (understanding mistake) | https://math.stackexchange.com/questions/1737564/conditional-probability-question-understanding-mistake | <p>I'm trying to understand the following question:</p>
<blockquote>
<p>An engineer conducts tests to find out if circuits of a certain type are prone to overheating. 30% of all such circuits are prone to overheating. If the circuit is prone to overheating, the test will report it is not prone to overheating with pr... | <p>Your method doesn't work because you have to find: </p>
<p>P(O | I on $1^{st}$ test $\cap$ P on $2^{nd}$ test), but you have calculated </p>
<p>What you have calculated is $P(O | \text{I on a test}) * P (O | \text{P on a test})$ which isn't the probability of a specific event.</p>
<p>The first part:</p>
<p>$P(O ... | 311 |
probability | Axiomatic probability intersection formula | https://math.stackexchange.com/questions/2978525/axiomatic-probability-intersection-formula | <p>I'm new to probability and I'm currently studying its axiomatic definition. I'm having a real hard time trying to understand the following exercise:</p>
<p>"
Tomorrow there is an exam. Esther has studied really hard, and she only has <span class="math-container">$\frac 1 5$</span> probability of not passing the ex... | <blockquote>
<p>My question is: How is that value achieved? How is it that the intersection of <span class="math-container">$\tfrac{1}{5}$</span> and <span class="math-container">$\tfrac{1}{3}$</span> equals <span class="math-container">$\tfrac{1}{8}$</span>?</p>
</blockquote>
<p>Well, <span class="math-container">... | 312 |
probability | Showing probability that $A$ and $B$ flip the same number of heads is equal to a total of $k$ heads. | https://math.stackexchange.com/questions/2978537/showing-probability-that-a-and-b-flip-the-same-number-of-heads-is-equal-to-a | <p>Question: A fair coin is independently flipped <span class="math-container">$n$</span> times, <span class="math-container">$k$</span> times by <span class="math-container">$A$</span> and <span class="math-container">$n − k$</span> times by <span class="math-container">$B$</span>. Show that the probability that
<spa... | <p>By symmetry we could assume <span class="math-container">$k \le n-k$</span>. The probability that <span class="math-container">$A$</span> and <span class="math-container">$B$</span> flip the same number of heads would be <span class="math-container">$\sum_{i=0}^{k}{\binom{k}{i}\binom{n-k}{i}(\frac{1}{2})^n} = (\frac... | 313 |
probability | What should be the general formula for the following permutation problem? | https://math.stackexchange.com/questions/1495199/what-should-be-the-general-formula-for-the-following-permutation-problem | <p>If $q$ number of elements are scheduled only to stay together, without having any specific order, what would be the permutation of $r$ elements taken from $n$ elements?</p>
<p>For example, suppose we have 5 alphabets $A, B, C, E, F$. If A and E always stay together, how many permutations are possible if we use 3 ch... | <p>Suppose out of n elements, q elements stay together, then we can consider a total of n - q elements and one additional group. Also, as those q elements can further have different permutations, we have to take additional factor of q! multiplied with the terms which include that group in them and as per your statement... | 314 |
probability | How to prove that $P(A) + P\left( {\bar A} \right)P\left( {B|\bar A} \right) = 1 - P\left( {\bar A \cap \bar B} \right)$? | https://math.stackexchange.com/questions/3832297/how-to-prove-that-pa-p-left-bar-a-rightp-left-b-bar-a-right-1 | <p>How to show that without using Venn Diagram</p>
<p><span class="math-container">$P(A) + P\left( {\bar A} \right)P\left( {B|\bar A} \right) = 1 - P\left( {\bar A \cap \bar B} \right)$</span> ?</p>
<p>Effort so far</p>
<p><span class="math-container">$P(A) + P\left( {\bar A} \right)P\left( {B|\bar A} \right) = P(A) + ... | <p><span class="math-container">$$P(A) + P(\overline{A}) P(B\mid \overline{A})$$</span><span class="math-container">$$ = P(A) + P(\overline{A}) \frac{P(B\cap \overline{A})}{P(\overline{A})}$$</span>
<span class="math-container">$$= P(A) + P(B \cap \overline{A}) $$</span>
<span class="math-container">$$=P(A \cup (B \cap... | 315 |
probability | Probability question on sample space | https://math.stackexchange.com/questions/4044378/probability-question-on-sample-space | <p>Recently, I encountered a probability question which can be phrased differently:</p>
<p>Q1: A letter is chosen at random from the word <strong>MISSISSIPPI</strong>. What is the sample space.</p>
<p>Q2: The letters from the word <strong>MISSISSIPPI</strong> are put into a bag. What is the sample space.</p>
<p>Q3: Pic... | <p>For Q2, it's still the same sample space because the letters are still indistinguishable. If you pull an <code>S</code> at random from the bag (I'm imagining Scrabble tiles here), you still don't know whether it was the first <code>S</code> in <code>MISSISSIPPI</code> or one of the others.</p>
<p>If we let the rando... | 316 |
probability | At least K successes in n tries without replacement | https://math.stackexchange.com/questions/1680387/at-least-k-successes-in-n-tries-without-replacement | <p>A bit surprisingly I can't find the answer to exactly my question. I am looking for the formula to calculate at least k successes with n tries without replacement.</p>
<p>For example take the bag/balls problems. Let's say 250000 balls in the bag 250 white 249750 blue. If you draw 8500 balls what is the probability ... | <p>The count of favoured items in a sample selected from a population <em>without replacement</em> has a <strong>hypergeometric distribution</strong>.</p>
<p>When the population is size $N$ with $K$ favoured items, and the sample is of size $n$, then the count $W$ of favoured items in the sample having size $k$ has pr... | 317 |
probability | Proof of $P(A|B)=1−P(A^{c}|B)$ | https://math.stackexchange.com/questions/2484262/proof-of-pab-1%e2%88%92pacb | <p>I see that there is a "fact" $P(A|B)=1−P(A^{c}|B))$, this can be deduced or what is the intuition? I can see that the "domain" is reduced in both cases to $B$ and that we use $A$ and $A^{c}$ and this makes sense I just don't know where to take this "fact" from.</p>
| <p>$$\begin{align}\mathsf P(A^\complement\mid B) ={}& \dfrac{\mathsf P(A^\complement\cap B)}{\mathsf P(B)}&&\text{by definition of conditional probability}\\[1ex] ={}& \dfrac{\mathsf P(B)-\mathsf P(A\cap B)}{\mathsf P(B)} && \raise{2ex}{\text{via the Law for Total Probability}\\{\small \mathsf P... | 318 |
probability | 3 of a kind, 5 rolls, 6-sized dice | https://math.stackexchange.com/questions/3476371/3-of-a-kind-5-rolls-6-sized-dice | <p>I am reliably informed that the probability of getting 3 of a kind in 5 rolls of a 6-sized dice is approximately <span class="math-container">$0.1929$</span>.</p>
<p>I'm assuming this excludes 4-of a kind and 5-of a kind but not full-house (3 of a kind plus two of a kind).</p>
<p>Trying to check this I reasoned: G... | <p>You calculated the probability that the first three rolls are the same, but the last two rolls produce a different result. However, there are
<span class="math-container">$$\binom{5}{3} = 10$$</span>
sequences in which exactly three of the five rolls yield the same value. This is your missing factor of <span class... | 319 |
probability | Odds of winning plus odds of losing do not equal 100% ??? | https://math.stackexchange.com/questions/2980503/odds-of-winning-plus-odds-of-losing-do-not-equal-100 | <p>If i have four dice i calculate the chances of getting at least one 2 as 864 ÷1296 = 66,66% since if on one dice a 2 comes up then it does not matter what comes up on the other 3 dice (1×6×6×6 possible outcomes × 4 times = 864), i still have thrown a two.</p>
<p>If i now work out the chances of not throwing at leas... | <p>The probability of no dice showing a two is <span class="math-container">$\left(\frac{5}{6}\right)^4\approx 48.22\%$</span>. The probability of at least one dice showing a two consequently is <span class="math-container">$1-\left(\frac{5}{6}\right)^4\approx 51.77\%$</span>. You can also obtain this number combinator... | 320 |
probability | "Inherently independent" distributions | https://math.stackexchange.com/questions/4172926/inherently-independent-distributions | <p>Given two cdf's <span class="math-container">$F_1, F_2\colon [0,1]\to\mathbb{R}$</span>, it is always possible to find two real-valued random variables <span class="math-container">$X_1, X_2$</span> such that <span class="math-container">$X_i$</span> is distributed according to <span class="math-container">$F_i$</sp... | <p>If <span class="math-container">$F$</span> is a cdf let <span class="math-container">$G$</span> be the "inverse" <span class="math-container">$G(u):=\inf\{x: F(x)> u\}$</span>. If <span class="math-container">$U$</span> is uniformly distributed on <span class="math-container">$(0,1)$</span> then the cd... | 321 |
probability | One problem about the villages(Probability theory). | https://math.stackexchange.com/questions/4163648/one-problem-about-the-villagesprobability-theory | <p>The taxi driver drives through four villages <span class="math-container">$W,Z,X,Y.$</span> These roads form a quadrangle where <span class="math-container">$WX = 5, WZ = 10, ZY = 5$</span> and <span class="math-container">$XY = 10.$</span> With a probability of <span class="math-container">$\dfrac{1}{4},$</span> a ... | 322 | |
probability | What is the probability that at least two spinner land on Red | https://math.stackexchange.com/questions/4171461/what-is-the-probability-that-at-least-two-spinner-land-on-red | <p>Three spinners are marked with equal amounts of Red, Blue and Yellow. At a particular instance, all three are spun together. What is the probability that at least two of the spinners land on red?</p>
<p>The at least part is confusing me.</p>
<p>My attempt:
So if all three lands on red the probability will be: <span ... | <p>i) if two of them land on red - choose which two spinners land on red, the third can be either blue or yellow. So if <span class="math-container">$X$</span> is the event of a spinner landing on red,</p>
<p><span class="math-container">$ \displaystyle P(X=2) = {3 \choose 2} \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \... | 323 |
probability | Drawing without replacement. Problem formally specifying the probability space. | https://math.stackexchange.com/questions/4174914/drawing-without-replacement-problem-formally-specifying-the-probability-space | <p>An urn contains <span class="math-container">$4$</span> blue and <span class="math-container">$4$</span> red marbles. At first a marble is drawn (without looking) and removed from the urn. Then, a marble is drawn from the urn, its color recorded and put back in the bag. This process is repeated <span class="math-co... | 324 | |
probability | what is the Pr(X=x)? | https://math.stackexchange.com/questions/2159690/what-is-the-prx-x | <p>So this is an example straight from a book, like an example to help teach the material yet it makes absolutely no sense or I am just not seeing where they make the jump at.</p>
<p>We have the following: Consider the experiment consisting of 2 rolls of a fair 4-sided die. Let X be a random variable, equal to the max... | <p>For example, $X=3$ consists of the sample points $(1,3), (2,3), (3,1), (3,2), (3,3)$. There are $5$ of them, while the sample space has $16$ points, all equally probable, so $P(X=3) = 5/16$.</p>
| 325 |
probability | Coin tossing - what's more probable? | https://math.stackexchange.com/questions/4174777/coin-tossing-whats-more-probable | <p>I am solving the following probability exercise. The solution I have found is very counter intuitive and I feel It is wrong, but I can't seem to understand why.</p>
<p>A fair coin is tossed twice, you have to decide wheter it is more likely that two heads
showed up given that: 1) at least one toss is head, 2) the se... | <p>It is correct.</p>
<p>There are four equally likely outcomes (HH, HT, TH, TT) of which one outcome has two heads (HH).</p>
<p>In question (1) there are two other possibilities (HT, TH).</p>
<p>In question (2) there is only one other possibility (TH). This makes HH conditionally more likely by excluding consideratio... | 326 |
probability | $X\stackrel d= Y$. What does it exactly mean? | https://math.stackexchange.com/questions/4086903/x-stackrel-d-y-what-does-it-exactly-mean | <p>When we write <span class="math-container">$X\stackrel d= Y$</span>, does this mean that <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> have exactly the same distribution?</p>
<p>For example, <span class="math-container">$X\sim\mathcal N(\mu,\sigma^{2}) \ \text{and}\ X\stackrel d... | <p><span class="math-container">$X\overset{d}{=}Y$</span> does mean equality in distribution, which implies that the distribution functions of <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> agree, that is,
<span class="math-container">$$
F_X(t)=F_Y(t),\quad\forall t\in\Bbb R.
$$</sp... | 327 |
probability | Expected value of the exponential of a normal-lognormal mixture in the special case where the correlation between the normal and lognormal r.v.s is 1 | https://math.stackexchange.com/questions/4176572/expected-value-of-the-exponential-of-a-normal-lognormal-mixture-in-the-special-c | <p>Is there any way of analytically determining the expected value of <span class="math-container">$Z=e^{\alpha X e^{\beta X}}$</span>, where <span class="math-container">$X\sim\mathcal N(0,1)$</span> and <span class="math-container">$\alpha,\beta$</span> are known constants?</p>
<p><strong>A couple of the methods I tr... | 328 | |
probability | Basic probability of at least one independent event happening | https://math.stackexchange.com/questions/2988464/basic-probability-of-at-least-one-independent-event-happening | <p>If four people are in a room for 1 hour, each on their own very old laptop and each laptop has a 10% chance of crashing during that time, then I thought the probability of at least one laptop crashing would be:</p>
<p>1/10 + 1/10 + 1/10 + 1/10 = 2/5</p>
<p>But then if the chance of crashing was 25% it would then b... | <p>It is just a binomial distribution where <span class="math-container">$n=4,p=\frac{1}{10}$</span></p>
<p><span class="math-container">$P[n,k]=\binom{n}{k}*p^k*(1-p)^{n-k}$</span></p>
<p><span class="math-container">$P[4,1]=\binom{4}{1}*(0.1)^1*(0.9)^3=0.2916$</span></p>
<p><span class="math-container">$P[4,2]=\bi... | 329 |
probability | probability - picking up from a box | https://math.stackexchange.com/questions/4177140/probability-picking-up-from-a-box | <p>I have the following question, but I fail to get the right answer. There are two boxes - box <span class="math-container">$A$</span> and box <span class="math-container">$B$</span>. Box <span class="math-container">$A$</span> has <span class="math-container">$5$</span> red balls and <span class="math-container">$3$<... | <p>You have to attack the problem in stages.</p>
<p>First, you have to calculate <span class="math-container">$f(a), f(b)$</span> where</p>
<p><span class="math-container">$f(a) =$</span> probability that balls are being drawn from box A and</p>
<p><span class="math-container">$f(b) =$</span> probability that balls are... | 330 |
probability | What is the probability that you have more than 3 deaths in an infected family of 6 if each pair of people have a different probability of death? | https://math.stackexchange.com/questions/4177704/what-is-the-probability-that-you-have-more-than-3-deaths-in-an-infected-family-o | <p>If say 2 people individually had a .05 chance of survival, 2 individually had a .10 chance of survival, and 2 individually had a .20 chance of survival (pulled these numbers out of a hat)</p>
<p>What is the chance that the family has more than 3 deaths.</p>
<p>The method I have is quite tedious and I was wondering ... | <p>Kinda morbid mate. Nevertheless. If the chances of survival are <span class="math-container">$p_i$</span> for <span class="math-container">$i=1,2,\dots,6$</span>, then the chance of</p>
<ul>
<li>only the <span class="math-container">$i$</span>-th person surviving is <span class="math-container">$p_i\prod_{i\neq j}(1... | 331 |
probability | Probability each table leg was in each spot. | https://math.stackexchange.com/questions/3379150/probability-each-table-leg-was-in-each-spot | <p>I have a fold up table at home with six legs and three areas. Each area takes 2 legs to keep the table up, but when stored three legs are positioned on the left and three on the right.</p>
<p>Everytime I setup the table I randomly take the three legs on the left, put two on the left side and one in the middle. Then... | <p>Suppose the legs are numbered and ordered in a list, for example [1,2,3,4,5,6]. When packed away the first three legs in the list are on the left, the last three are on the right. When erected the first two are on the left, the middle two in the middle and the last two on the right. </p>
<p>Let's consider the possi... | 332 |
probability | Conditional probability - a system with 3 parts | https://math.stackexchange.com/questions/4180245/conditional-probability-a-system-with-3-parts | <p>I fail to see what I have done wrong solving the following problem:
Consider a system with 3 parts A,B,C. Part A works with probability 0.8, part B with 0.8 and part C with 0.9 (they are independent). The system is considered to work only if there are at least two parts working. I wish to calculate what the probabil... | <p>Your mistake is in the numerator, in finding probability that the system does not work but part <span class="math-container">$1$</span> works (I am calling it part <span class="math-container">$1$</span> instead of part <span class="math-container">$A$</span>).</p>
<p>If <span class="math-container">$B$</span> is th... | 333 |
probability | Classical Probability and Combinatorics | https://math.stackexchange.com/questions/1415911/classical-probability-and-combinatorics | <p>Shuffle a standard deck of cards and cut it into three piles. What is the probability that a face card will turn up on top of one of the piles? </p>
<p>There are 12 face cards (four jacks, four queens and four kings) in the deck.</p>
| <p>As has been mentioned in the comments, the split into three decks is just a red herring - the problem is equivalent to just picking the three top cards assuming the deck has been shuffled well.</p>
<p>The probability of getting at least one court card is equal to one minus the probability of the inverse case: getti... | 334 |
probability | Probability and marbles | https://math.stackexchange.com/questions/3397057/probability-and-marbles | <p>My brother brings a certain number of his marbles to play with in my room. Each marble is distinct. He has 8 total marbles that are either red or blue. One day, I spotted two red marbles in my room. The probability that any two of his marbles (of those that he plays in my room), randomly chosen, both being red is 1/... | <p>Assume your brother brings <span class="math-container">$x+2$</span> red and <span class="math-container">$y$</span> blue marbles, and leaves <span class="math-container">$2$</span> marbles, where <span class="math-container">$0\leq x+y\leq 6$</span>.</p>
<p>Then the probability that those two marbles are both red ... | 335 |
probability | genetic combination exercise | https://math.stackexchange.com/questions/4139143/genetic-combination-exercise | <p>Consider families with two children, in which both parents have been identified as carriers of an autosomal recessive allele (Aa). At least one of the children shows the corresponding phenotype. When adding all the children of such families, what proportion of them will show this phenotype?</p>
<p>Why is the correct... | <pre><code> A a
A AA Aa
a Aa aa
</code></pre>
<p>Based on the Punnett square above there is a <span class="math-container">$1/4$</span> chance that a child will show the phenotype. Let <span class="math-container">$X$</span> be the number of children in the family of <span class="math-container">$2$</span> that show ... | 336 |
probability | Law of total expectation - a toss of a coin | https://math.stackexchange.com/questions/4182140/law-of-total-expectation-a-toss-of-a-coin | <p>There is the following question: There is a coin with probability <span class="math-container">$p$</span> to be <span class="math-container">$H$</span> and <span class="math-container">$q$</span> to be <span class="math-container">$T$</span>. I'm asked what is the expectation of the tossing number, considering I tos... | 337 | |
probability | 1.96 and the Standard Normal Distribution | https://math.stackexchange.com/questions/3336298/1-96-and-the-standard-normal-distribution | <p>I am interested in learning deeper about the number 1.96 used in the test of 95% confidence with a normal distribution. </p>
<p>More specifically, I am interested in whether someone could provide a numerical example of this, and how 1.96 is calculated using the 97.5th percentile, or anybody knows somewhere where it... | <p><span class="math-container">$$X \sim N(\mu,\sigma^2)$$</span></p>
<p><span class="math-container">$$P \bigg( \mu - 1.96\sigma < X < \mu + 1.96\sigma\bigg) = 0.95$$</span></p>
<p><span class="math-container">$$P\bigg( \mu - 1.96\sigma > X\bigg) = 0.025$$</span></p>
<p><span class="math-container">$$P\big... | 338 |
probability | Biased coins that are uniformly distributed | https://math.stackexchange.com/questions/358904/biased-coins-that-are-uniformly-distributed | <p>I have a bag that contains coins, these coins could be biased coins, and each coin has a certain pre-determined probability of head/tail (independently of the other coins). This pre-determined probability is derived from a uniform distribution over <span class="math-container">$[0,1].$</span></p>
<p>I draw a coin a... | <p><strong>Hint:</strong> Compute the expected number of flips needed to get a tail using a single biased coin whose probability of producing a head is $p$.</p>
| 339 |
probability | Flip a biased coin until a head appears | https://math.stackexchange.com/questions/1216842/flip-a-biased-coin-until-a-head-appears | <p>I'm having some trouble with this problem</p>
<blockquote>
<p>Suppose you flip a biased coin until a head appears. The coin has a $75%$ chance of coming up tails. Let $n$ be the number of flips that you need to do. What is the probability of the following events:</p>
<p>a) $n$ is at most $3$?</p>
<p>b) ... | <p>The biased coin in your case only means that $P(T)=\frac{3}{4}$.
<br>First note that to stop at a single Head in $n$ tosses, you need to get Tail in the first $n-1$ tosses and a Head in the last (that is $n^{th}$ toss).</p>
<p>$1.$ When $n\leq3$.
You can get a head in these ways - $H,TH,TTH$.
<br>(Here $TH$ implies... | 340 |
probability | What is the probability we picked a fake coin if we have 7 proper and 1 with heads on both sides if after we flipped it 7 times we got heads always | https://math.stackexchange.com/questions/4183975/what-is-the-probability-we-picked-a-fake-coin-if-we-have-7-proper-and-1-with-hea | <p>So, I'm going to reformulate the problem because I think I did a bad job in the title because of the word limit:<br/>
We have 8 coins in a box - 1 has "heads" on both sides and the remaining 7 are normal. We pick 1 coin randomly and toss it 7 times. If we got "heads" all 7 times, what is the prob... | <p>This can be solved with Bayes' Theorem.</p>
<p>The probability of having picked a fake coin <span class="math-container">$\displaystyle P(f) = \frac 18$</span>. The probability of having picked a real coin <span class="math-container">$\displaystyle P(r) = \frac 78$</span>.</p>
<p>In the event you had picked a fake ... | 341 |
probability | What is the probability that a random family of 4 gets wiped out by Thanos? (i.e 50 % of population being dead) | https://math.stackexchange.com/questions/2875086/what-is-the-probability-that-a-random-family-of-4-gets-wiped-out-by-thanos-i-e | <p>I am confused about this. Do we need to consider in the factor of household distribution in the world?</p>
<p>The context is the movie <em>Avengers: Infinity War.</em> The important part of the question is a movie spoiler:</p>
<blockquote class="spoiler">
<p> At the end of the movie, Thanos gets all six infinity... | <p>Consider each person getting turned to dust as an independent event, which happens with probability $1/2$. Then the probability of an entire family of four getting it is $(1/2)^4 = 1/16.$</p>
<p>And to answer your question about the household disbribution, if half of the family <em>has</em> to get it, then you alre... | 342 |
probability | Probability that two plates with numbers ranging from $0001-9999$ have at least two common digits in them. | https://math.stackexchange.com/questions/3502794/probability-that-two-plates-with-numbers-ranging-from-0001-9999-have-at-least | <p>I would calculate it as this <span class="math-container">$\frac{10^4-9^4-9^4+8^4}{10^4}$</span>. But it may be incorrect Because I summing the <span class="math-container">$10^4-9^4$</span> and <span class="math-container">$9^4-8^4$</span>. Is it the correct? </p>
| <p>Let's consider "common digits" to mean the digit appears in both plate numbers in any order. Let's find the probability that the two plates have no common digit.</p>
<p>If the first plate has one distinct digit, this can happen in any of 9 ways (<span class="math-container">$0000$</span> is not possible), so there ... | 343 |
probability | An urn of 4 balls with 2 colors. Pick 2 balls and place them back 4 times. What's the probability of picking 2 balls of the same color twice in a row? | https://math.stackexchange.com/questions/3509877/an-urn-of-4-balls-with-2-colors-pick-2-balls-and-place-them-back-4-times-what | <p>An urn of 4 balls with 2 colors. Pick 2 balls and place them back 4 times. What's the probability of picking 2 balls of the same color twice in a row?</p>
<p>So the probability of picking 2 balls of the same color is <span class="math-container">$2\choose1$$2\choose2$</span>/<span class="math-container">$4\choose2$... | <p>Assumptions: </p>
<ol>
<li>There are two balls of one color and two balls of a second color</li>
<li>Suppose the colors are Green and Red. A successful outcome occurs in any of these cases when you choose two green followed by two green, two green followed by two red, two red followed by two green, or two red follo... | 344 |
probability | Probability of trifecta box bet | https://math.stackexchange.com/questions/3534016/probability-of-trifecta-box-bet | <p>In a horse race there are 10 horses. Bob wants to make a "trifecta Box bet". A trifecta box bet is when you choose the first three horses that finish the race in ANY order.
What is the probability to win a single trifecta box bet assuming every horse has equal chances to win.</p>
<p>My solution:
<span class="math-... | <p>JMoravitz showed that you can use your probability space as all possible trifectas (ignoring specific orders of the horses). You attempted the problem by setting the probability space as all possible orders of the horses. For the numerator, you chose all possible ways of choosing three horses to finish in the first ... | 345 |
probability | Probability of picking out two heads in a bag of four coins? | https://math.stackexchange.com/questions/3536847/probability-of-picking-out-two-heads-in-a-bag-of-four-coins | <p>All fair coins.</p>
<p>You pick 2 out of the bag and look at them, they are all heads.</p>
<p>So what is the probability of it?</p>
<p>I made a table under but it does not seem to work.</p>
<pre><code>A = Coin #1, B = Coin #2, C = Coin #3, D = Coin #4.
A B C D
1 H H H H >>... | <p>There are six possible combinations of coins:
<span class="math-container">$$AB,AC,AD,BC,BD,CD$$</span></p>
<p>For each row, you need to check if both of them are heads or not. It breaks down to this:</p>
<p><span class="math-container">$$\begin{array}{c|c}\text{Row} & \text{Number Pairs With Two Heads} \\ \hl... | 346 |
probability | What is the probability of winning first prize, given the following information? | https://math.stackexchange.com/questions/3546985/what-is-the-probability-of-winning-first-prize-given-the-following-information | <p>A weekly lottery consists of 3 numbers drawn from the digits 0 through 9 with no repetition of digits. The first prize goes to the person with the correct sequence. Second prizes go to people with the correct digits in some other sequence. You buy a ticket.</p>
<p>a) What is the probability of winning the first pri... | <p>a) <span class="math-container">$$\dfrac{1}{_{10}P_3} = \dfrac{(10-3)!}{10!} = \dfrac{1}{720}$$</span></p>
<p>b) <span class="math-container">$$\dfrac{3!-1}{_{10}P_3} = \dfrac{(3!-1)(10-3)!}{10!} = \dfrac{1}{_{10}C_3} - \dfrac{1}{_{10}P_3} = \dfrac{5}{720}$$</span></p>
<p>c) Because winning in the past is independ... | 347 |
probability | A die is thrown 1000 times. Find the limits within which the number of eyes coming up will lie with probability greater than 0.99. | https://math.stackexchange.com/questions/3601681/a-die-is-thrown-1000-times-find-the-limits-within-which-the-number-of-eyes-comi | <p>Friends, I found trouble understanding this sentence. This is an exercise from a homework on The Central Limit Theorem. Can someone explain what this question is trying to ask? Much thanks. </p>
| <p>Let <span class="math-container">$X_i$</span> be the number of eyes showing on the <span class="math-container">$i$</span>-th roll. Then:</p>
<p><span class="math-container">$$X = \sum_{i=1}^{1000}X_i$$</span></p>
<p><span class="math-container">$$E[X] = \sum_{i=1}^{1000}E[X_i] = 1000E[X_i] = 3500$$</span></p>
<p... | 348 |
probability | Probability of at least 1 person in Group A being in a group of X size with a general population | https://math.stackexchange.com/questions/4047746/probability-of-at-least-1-person-in-group-a-being-in-a-group-of-x-size-with-a-ge | <p>im trying to find a mathematical way to calculate the percentage chance that there is at least 1 Cheater in any given match chosen at random.</p>
<p>Game has 100 players per match</p>
<p>Total Players (including the Cheaters): 3000</p>
<p>there are 3 scenarios needing to be tested, a group of 500, 100, 25 Cheaters</... | <p>I will use the following notation.</p>
<p>Population of Players: <span class="math-container">$P$</span></p>
<p>Number of Cheaters: <span class="math-container">$C$</span></p>
<p>Number of players in a match: <span class="math-container">$M$</span></p>
<p>The probability of at least one cheater in a match is <span c... | 349 |
probability | Probability of throwing a die | https://math.stackexchange.com/questions/4137950/probability-of-throwing-a-die | <p><strong>Question</strong>: How many times should we throw a die if we want that the sum of points obtained
was at least 4500 with probability <span class="math-container">$p \geq 0; 975?$</span>
(use the central limit theorem).</p>
<p>I know that the probability of getting a given value for the total on the dice ma... | <p>Outline:</p>
<p>For a single roll of the die the mean number of
points is <span class="math-container">$\mu = 3.5$</span> and the variance
of the number of points is <span class="math-container">$\sigma^2 = 2.916667.$</span>
These values can be found using the definitions of the mean and variance of a discrete rando... | 350 |
probability | By means of an example, show that $P(A) + P(B) = 1$ does not mean that $B$ is the complement of $A$ | https://math.stackexchange.com/questions/3160157/by-means-of-an-example-show-that-pa-pb-1-does-not-mean-that-b-is-th | <p>I'm in grade 10, and I've just started to learn about complementary events. I am rather perplexed with this question. Isn't this question kinda contradictory, since <span class="math-container">$P(A) + P(A') = 1$</span>?</p>
<p>This is what I got to:</p>
<p><span class="math-container">$P(A) + P(B) = 1$</span></p>... | <p>Take any event of probability <span class="math-container">$\frac 1 2$</span> and take <span class="math-container">$B=A$</span>.</p>
| 351 |
probability | Something unclear about central limit theorem / law of large numbers | https://math.stackexchange.com/questions/4187056/something-unclear-about-central-limit-theorem-law-of-large-numbers | <p>Say I have <span class="math-container">$S_n = X_1 + X_2 +...+X_n$</span> where <span class="math-container">$X_i \sim Ber(1/2)$</span>. Then:
<span class="math-container">$$\lim_{n\to\infty}P(|S_n-\frac{n}{2}|<\frac{\sqrt n}{2})\implies \lim_{n\to\infty} P(|M_n-\frac{1}{2}|<\frac{1}{2\sqrt n}) \to 0$$</span>
... | <p>The law of large numbers tells you that
<span class="math-container">$$\lim_{n\to\infty} M_n-1/2=0$$</span> almost surely, i.e.,
<span class="math-container">$$\mathsf P\left(\lim_{n\to\infty} M_n-1/2=0\right)=1,$$</span>
but this does not imply that <span class="math-container">$$\lim_{n\to\infty}\mathsf P\left(\lv... | 352 |
probability | The Executioner Conundrum | https://math.stackexchange.com/questions/1832020/the-executioner-conundrum | <p>You are a military executioner tasked with eliminating some of the most dangerous criminals on Earth. You are handed 100 such criminals for immediate termination. However, just as you are about to execute them, word comes from a highly reliable source that 1 of the 100 is not a criminal at all. In fact he is one of ... | <p>Let's say you pick $n$ people out of $100$. There is a $\frac{n}{100}$ probability that the good person is in the set of $n$ people and thus a $\frac{n}{100}$ probability that the good person will be killed.</p>
<p>This is a classic example of a very long word problem with lots of extraneous information that has an... | 353 |
probability | Definition of almost sure convergence from Casella & Berger | https://math.stackexchange.com/questions/4187882/definition-of-almost-sure-convergence-from-casella-berger | <p>In Casella and Berger, the definition of almost sure convergence of <span class="math-container">$\{X_n\}$</span> to <span class="math-container">$X$</span> is</p>
<p><span class="math-container">$$P(\lim\limits_{n \rightarrow \infty}|X_n - X| < \epsilon) = 1$$</span></p>
<p>for all <span class="math-container">$... | 354 | |
probability | A question related to probability | https://math.stackexchange.com/questions/3506173/a-question-related-to-probability | <p>If I have a coin then chances of getting a <span class="math-container">$head$</span> or a <span class="math-container">$tail$</span> is <span class="math-container">$50-50$</span>. But why don't we take in account the case where coin is neither head and tail, where coin is standing upright, or it is making an angle... | <p>Usually, when mathematicians talk about 'a coin flip' they mean the idea of an ideal coin flip, where the probability of getting 'head' or 'tails' is exactly one half for either of the possibilities. This may or may not be a good model for the outcomes of a real coin flip. But there is nothing stopping you from thin... | 355 |
probability | How can I find the probability of waking up at a precise minute? | https://math.stackexchange.com/questions/2080754/how-can-i-find-the-probability-of-waking-up-at-a-precise-minute | <p>How can I find the probability of waking up at a precise minute? Say I fall asleep at $10$ PM and wake up at $6:01$ AM. There's a total of $481$ minutes we are dealing with so the odds of waking up at an exact minute would be $1:480$ right? And the probability would be $0.2$% ($1/481$), correct? But what about exter... | <p>For the average British population, the duration of the sleep is well modeled by a Gaussian with $\mu=7.04$h and $\sigma=1.55$h [<a href="https://dx.doi.org/10.1111/j.1365-2869.2004.00418.x" rel="nofollow noreferrer">Groeger et al. 2004</a>]. To estimate the probability to wake-up after $m$ minutes,</p>
<p>$$W_m:=\... | 356 |
probability | Probability of the next random number based on previous numbers | https://math.stackexchange.com/questions/2082137/probability-of-the-next-random-number-based-on-previous-numbers | <p>A random number generator generates a number between 0-9. Single digit, totally random. </p>
<p>We have the list of previous digits generated. </p>
<p>I would like to calculate what is the probability for each number between 0-9 to be the next number generated. </p>
<p>So we have something like: 0,2,3,4,6,4,9,1,3... | <p>Assuming this is a uniform distribution, each number is equally likely. That is, $\Pr(X=n) = \frac1{10}$, for $n=0,1,2,\ldots,9$.</p>
<p>The expected value for the next number, regardless of the history, is always
$$E[X] = 0(\tfrac1{10}) + 1(\tfrac1{10}) + \cdots + 9(\tfrac1{10}) = 4.5$$</p>
<p>although that's pre... | 357 |
probability | Example of function of a single random variable. | https://math.stackexchange.com/questions/2932875/example-of-function-of-a-single-random-variable | <p>Take a look at this document: <a href="http://hal.in2p3.fr/in2p3-01082914v2/document" rel="nofollow noreferrer"><em>Functions of random variables</em>; Abdel-Hamid Soubra, Emilio Bastidas-Arteaga</a></p>
<p>In the section <strong><span class="math-container">$2.2$</span></strong>, they have given an example about a... | <p>The function used in the book is precisely a linear one, written</p>
<p><span class="math-container">$$y=\frac{x-\mu}\sigma$$</span> or <span class="math-container">$$x=\sigma y+\mu.$$</span></p>
<p>If you plug this in the initial distribution,</p>
<p><span class="math-container">$$f_X(x)=\frac{1}{\sigma \sqrt{2 ... | 358 |
probability | What independent events actually means? | https://math.stackexchange.com/questions/2795511/what-independent-events-actually-means | <blockquote>
<p>A bin has $2$ balls, one is black and one is white. Every round a uniformly chosen ball is drawn from the bin. If the color of the ball is white, then the ball is returned to the bin with an additional white ball. If the ball is black the experiment is over. Let $X$ be the number of rounds in the abov... | <p>A difficulty encountered by your argument is that you do not have a clear definition of $X_j$ from the problem statement.</p>
<p>You seem to be tempted to say $X_j = 0$ if the experiment ends before the $j$th drawing.
But if there is no $j$th drawing, is "the number of white balls in the $j$th drawing" even defined... | 359 |
probability | What is the probability of winning Hearthstone's heroic tavern brawl? | https://math.stackexchange.com/questions/1972993/what-is-the-probability-of-winning-hearthstones-heroic-tavern-brawl | <p>Link for the description: <a href="http://us.battle.net/hearthstone/en/blog/20324471/introducing-heroic-tavern-brawl-10-17-2016" rel="nofollow">http://us.battle.net/hearthstone/en/blog/20324471/introducing-heroic-tavern-brawl-10-17-2016</a></p>
<p>Lest assume that you build a very good deck with 60% winrate. </p>
... | <p>Let us make the following simplification: We <strong>continue</strong> to play games <strong>after</strong> reaching the twelve-win maximum or the three loss maximum.</p>
<p>Recognize then that to have reached twelve wins, then in the first fourteen games you play you will have lost at most two times.</p>
<p>$\bi... | 360 |
probability | Confusion in identifying independent events | https://math.stackexchange.com/questions/4190549/confusion-in-identifying-independent-events | <p>There are 6 white beads and 5 black beads in your pocket. You randomly pull the beads one by one out of your pocket and place them on a table. Probability that the third bead drawn is the first white.</p>
<p>Now the solution is : the prob.of drawning 1st black bead (5÷11) × the prob.of drawing 2nd black bead.(4÷10) ... | <p>Let <span class="math-container">$E_1$</span> denote the event that the 1st bead drawn is black.</p>
<p>Let <span class="math-container">$E_2$</span> denote the event that the 2nd bead drawn is black.</p>
<p>Let <span class="math-container">$E_3$</span> denote the event that the 3rd bead drawn is white.</p>
<p><span... | 361 |
probability | Probability that First $s$ Heads in a Row Occurs After $n$ Flips | https://math.stackexchange.com/questions/3022791/probability-that-first-s-heads-in-a-row-occurs-after-n-flips | <p>Flip a coin repeatedly. Let <span class="math-container">$E_s$</span> be the number of coin flips it takes before seeing <span class="math-container">$s$</span> heads in a row. What is <span class="math-container">$P(E_s=n)$</span>? Specifically, I am concerned with <span class="math-container">$P(E_4=E_3+k)$</span>... | <p>EDIT: The answer below isn't really an answer, as it failed to address the "in a row" component. See the comments for details.</p>
<p>So there are a total of <span class="math-container">$2^n$</span> possible sequences of flipping a fair coin <span class="math-container">$n$</span> times, and of those there are <sp... | 362 |
probability | Probability of two people from two different countries meeting in a different country and meeting each other | https://math.stackexchange.com/questions/2898661/probability-of-two-people-from-two-different-countries-meeting-in-a-different-co | <p>So I would like to know the probability of the following scenario to happen.</p>
<p>I am from India. Jane(say) is from Brazil. I moved to Canada to work four months ago and have been hopping from one airbnb to another. Jane moves from Brazil to Canada to study and happens to be in the room next to me in the same ai... | <p>I will try to answer the question I suspect is behind your question.</p>
<p>The probability of that sequence of coincidences if specified in advance of the observation that they happened is extremely small. Estimating it would require lots of assumptions I wouldn't even try to specify.</p>
<p>That's because you di... | 363 |
probability | Probability of Random Numbers in a Table Summing to $10$ | https://math.stackexchange.com/questions/4156924/probability-of-random-numbers-in-a-table-summing-to-10 | <p>Assume a table with dimensions <span class="math-container">$n$</span>x<span class="math-container">$n$</span>. In each of the <span class="math-container">$n^2$</span> spaces, a random number (<span class="math-container">$m$</span>) such that <span class="math-container">$m\in\mathbb{N}$</span> and <span class="ma... | <p>Q1: As I understood, you're asking of the probability that in the table that we get every two neighbours (vertically or horizontally) will add up to 10.</p>
<p>If (1,1) contains a number <span class="math-container">$a$</span>, then (1,2) and (2,1) must contain <span class="math-container">$(10-a)$</span>, then (1,3... | 364 |
probability | How would someone go about calculating the probability of the unlikely scenario where to friends meet on their vacation? | https://math.stackexchange.com/questions/4191840/how-would-someone-go-about-calculating-the-probability-of-the-unlikely-scenario | <p>Imagine this. Robert goes on vacation. Upon his arrival at the destination, he is unexpectedly greeted by a good friend of him, Jeremy. Weirdly enough, they happened to go on the same trip, be neighbors at the site, at the same time and place, and for the same duration. Keep in mind, none of them knew of the other's... | 365 | |
probability | What are the odds of sitting next to the same person on two flights? | https://math.stackexchange.com/questions/2903649/what-are-the-odds-of-sitting-next-to-the-same-person-on-two-flights | <p>My wife left on a business trip this morning. 20 people from the same company caught two consecutive flights. Each person checked in independently, yet my wife ended up sitting next to the same colleague on both flights!</p>
<p>What are the odds?</p>
<p>Assume both aeroplanes had 150 seats, in 3+3 configuration, i... | <p>Your wife has probability $\frac23$ to sit in a chair that has only $1$ chair next to it, and has probability $\frac13$ to sit in a chair that has $2$ other chairs next to it. </p>
<p>The probability that at the first flight your wife had no colleagues sitting next to her is:$$p_0:=\frac23\frac{\binom{148}{19}\bino... | 366 |
probability | Solving conditional probability without formula | https://math.stackexchange.com/questions/2904005/solving-conditional-probability-without-formula | <p>How can you solve conditional probability without formula - simply by logic and intuition?</p>
<p>For example, this problem has been circulated here and we all know the formal way to do it. Could anyone show how to logically solve it?</p>
<p>At a workplace 1% of the staff where injured during a year. 60% of all in... | <p>Women make up $30\%$ of the workforce, but sustain $40\%$ of the injuries. So women get injured at a higher rate.</p>
| 367 |
probability | 1 in 8 chance of an event decreases by 25%, what is it? | https://math.stackexchange.com/questions/2907512/1-in-8-chance-of-an-event-decreases-by-25-what-is-it | <p>If there is a 1 in 8 chance of an event and there is a further 25% reduction in this event happening what is the answer expressed in terms of 1 in X chance?</p>
<p>My first calculation I worked out as 1 in 32.
0.125 x 0.25 =0.03125</p>
<p>Then 1 in 12 (more guesswork) and then 3 in 32
0.125 x 0.25 = 0.03125
0.12... | <p>The probability starts out as $\frac 18$. If we reduce that by $25\%$ we multiply it by $\frac 34,$ getting a probability of $\frac 3{32}$. If you want to express that as $1$ in something it is $1$ in $\frac {32}3=10\frac 23$. There is no whole number answer.</p>
| 368 |
probability | Distribution of white caramels thrown away from a bag of white and black caramels (probability problem) | https://math.stackexchange.com/questions/2907888/distribution-of-white-caramels-thrown-away-from-a-bag-of-white-and-black-caramel | <blockquote>
<p>Suppose we have a bag containing $m$ white and $n$ black caramels. We pic a caramel and if it is white, we eat it, otherwise we put it back in the bag. If we take out $r$ black caramels succesively, then we believe that we have eaten all the white caramels and we throw the bag. What is the distributio... | <p>The probability that all of white caramels will be thrown away is $(\frac{n}{m+n})^r$.</p>
<p>The probability that exactly one will be eaten is $(1-(\frac{n}{m+n})^r)(\frac{n}{m+n-1})^r.$</p>
<p>The probability that exactly two will be eaten is $(1-(\frac{n}{m+n})^r)(1-(\frac{n}{m+n-1})^r)(\frac{n}{m+n-2})^r.$</p>... | 369 |
probability | Find the pdf of $Z=X+Y$ | https://math.stackexchange.com/questions/2908104/find-the-pdf-of-z-xy | <p>Let X and Y be jointly continuous with joint probability density function<br>
$$f_{X,Y}(x,y)=\frac{1}{x},0\leq y\le x\le1$$<br>
Find the pdf of $Z=X+Y$ </p>
<p>Here is my solution:<br>
$$F_Z(z)=P(Z\leq z)=P(X+Y\leq z)$$<br>
$$=\int_{-\infty}^{\infty}P(X+Y\leq z|X=x)f_X(x)dx$$<br>
$$=\int_{-\infty}^{\infty}P(Y\leq ... | <p>Notice, $x,y$ are <em>not</em> independent, so stick with the joint functions. Use the Jaccobian transformation.</p>
<p>Always keep your eye on the supports.</p>
<p>$$\begin{split}f_Z(z) &=\int_\Bbb R f_{X,Z}(x,z)\mathsf d x\\ &= \int_\Bbb R f_{X,Y}(x,z-x) \begin{Vmatrix}\dfrac{\partial (x,z-x)}{\partial ... | 370 |
probability | What is the probability that he gets at least one mail in each day? | https://math.stackexchange.com/questions/2912079/what-is-the-probability-that-he-gets-at-least-one-mail-in-each-day | <p>If someone gets $13$ mails over the period of $5$ weekdays. What is the probability that he gets at least one mail in each day?</p>
| <p>HINT - I would say:</p>
<p>If number of solutions of the equation $i_1+i_2+i_3+i_4+i_5 = 13$</p>
<ul>
<li><p>where $i_1,i_2,i_3,i_4,i_5\in (1,2,3,\cdots,13) = \omega$,</p></li>
<li><p>where $i_1,i_2,i_3,i_4,i_5\in (0,1,2,3,\cdots,13) = \Omega$</p></li>
</ul>
<p>then the sought probability $P=\frac{\omega}{\Omega}... | 371 |
probability | Is it true, that the probability for both events is always equal? If yes how to prove it, if not, why not? | https://math.stackexchange.com/questions/2913017/is-it-true-that-the-probability-for-both-events-is-always-equal-if-yes-how-to | <p>Imagine I have a real random variable $X$ with some distribution (continuous, discrete or continuous with atoms)</p>
<p>Now Imagine I have i.i.d. copies $X_1,...,X_n$, all independently and equally distributed as $X$</p>
<p>My claim is:</p>
<p>$$\mathbb{P}(X_2>X_1)=\mathbb{P}(X_2<X_1)$$
My secondy claim is ... | <p>Let $Y_n =\min(X_1,X_2 \cdots X_n)$, and let $y_n=\sum_{i=1}^n[X_i=Y_n]$ count the number of elements that attain that minimum.
Analogously, let $Z_n$ and $z_n$ be the maximum and maximum-count.</p>
<p>Then, by symmetry $P( X_{n} = Y_n \wedge y_n=1)=P(X_n=Y_n) P(y_n=1 \mid X_n=Y_n)=\frac{1}{n} P(y_n=1)$</p>
<p>The... | 372 |
probability | Two players pick cards from standard 52 card deck without replacement... | https://math.stackexchange.com/questions/2916365/two-players-pick-cards-from-standard-52-card-deck-without-replacement | <p>I am struggling with this interview prep question... SOS</p>
<p>Two players pick cards from standard 52 card deck without replacement: 1st player
picks a card, then 2nd, then again 1st, then 2nd etc. They stop once somebody picks a king (of any suit),
the player who picks the king wins. What is the probability that... | <p>In a given round of two draws, you start with $n$ cards of which $4$ are kings. The first player wins with probability $\frac 4n$. The second player wins with probability $\frac {n-4}n\cdot \frac 4{n-1}=\frac 4n\cdot\frac {n-4}{n-1}$ because they need the first player not to draw a king and there is one less card ... | 373 |
probability | Is the following probability question ambiguous? | https://math.stackexchange.com/questions/2920913/is-the-following-probability-question-ambiguous | <p>This is related to my previous question <a href="https://math.stackexchange.com/questions/2920891/find-the-number-of-ways-of-constructing-8-using-three-distinct-integers-from">here</a>.</p>
<blockquote>
<p>The numbers $0, 1, 2, 3, \ldots , 8$ are written on individual cards and placed in a bag. Three cards are ch... | <p>The problem is unambiguous. It does not matter whether your three numbers are chosen "all in one go" or "one-by-one," at the end of the day you have three random numbers either way. You are hung up on whether or not order matters. The thing is this: you can choose whether or not order matters when you do your comput... | 374 |
probability | Cat / mouse probability question | https://math.stackexchange.com/questions/2924838/cat-mouse-probability-question | <p>There exist 7 doors numbered in order from 1 to 7 (going from left to right). A mouse is initially placed at center door 4. The mouse can only move 1 door at a time to either adjacent door and does so, but is twice as likely to move to a lower numbered door than to a higher numbered door each time it moves 1 door.... | <p>I'll take a shot at this........</p>
<p>1) The eventual fateful outcome occurs when the total number of moves, lower versus higher, differs by 3.</p>
<p>So, a way to look at this as an <span class="math-container">$E(x) = n\cdot p$</span> type problem is to sum the <span class="math-container">$(n\cdot p)$</span>s... | 375 |
probability | Probability of Next Occurence | https://math.stackexchange.com/questions/2925412/probability-of-next-occurence | <p>I have collected data on the time duration between consecutive occurrences of a particular event ("success"), and the amount of time between consecutive "successes" (in days) seems to be distributed using a Gamma distribution. Intuitively, this is not a Bernoulli trial because the probability of a "success" in a gi... | 376 | |
probability | Entrance at gymnasium | https://math.stackexchange.com/questions/2929025/entrance-at-gymnasium | <p>Bill gave exams for the entrance at some specific gymnasium. <span class="math-container">$602$</span> students took part, which were classified, after the exams, in an ascending order, and the first <span class="math-container">$108$</span> students will be taken, which will accept to enter. Every student that has... | <p>This is a binomial distribution with <span class="math-container">$p=.02$</span>, <span class="math-container">$n=112$</span>, and five successes required. So the simple way to find the answer is simply to find a binomial calculator. For instance, <a href="https://stattrek.com/online-calculator/binomial.aspx" rel="n... | 377 |
probability | Density of a Random Variable | https://math.stackexchange.com/questions/2932126/density-of-a-random-variable | <blockquote>
<p>A point is chosen uniformly at random inside the triangle with vertices at <span class="math-container">$(0, 0), (0, 1)$</span> and <span class="math-container">$(1, 0)$</span>, meaning that the probability that the point lies in a certain region inside the triangle is proportional to the area of that... | <p>Let us work through this example together.</p>
<p>We want to find the relative size of the region of the triangle where <span class="math-container">$Z = max(X,Y)$</span> is less than equal some <span class="math-container">$a$</span>.</p>
<p>If <span class="math-container">$a\le 1/2$</span>, the region is a squar... | 378 |
probability | Probability of dividing boys in 2 groups | https://math.stackexchange.com/questions/2936889/probability-of-dividing-boys-in-2-groups | <p>A group of $2n$ boys is to be divided into two groups of $n$ boys . What is the probability that the two tallest boys are in different groups ? </p>
<p>This is how I attempted it: </p>
<p>The probability that the two boys are in same group can be obtained as follows:</p>
<p>First we separate those two particular ... | <p>Let us assume that the tallest boys are Andrew and Bruce. The configurations of this kind
<span class="math-container">$$ (A\text{ together with }n-1\text{ other people })\quad (B\text{ together with }n-1\text{ other people }) $$</span>
are <span class="math-container">$\binom{2n-2}{n-1}$</span> (it is enough to sel... | 379 |
probability | Probability of winning die game | https://math.stackexchange.com/questions/2937271/probability-of-winning-die-game | <p>Suppose we play a game where we roll a six-sided die. If a <span class="math-container">$4$</span>, <span class="math-container">$5$</span>, or <span class="math-container">$6$</span> is rolled, I get <span class="math-container">$1$</span> point. If a <span class="math-container">$1$</span>, <span class="math-conta... | <p>Well, sure one could take that approach, and consider all of the edge cases, and take large infinite sums over all of them, or one could make the following observation. (An observation that could make life a bit more easy is to realize that this dice game is equivalent to a coin flipping game, but even as so, the in... | 380 |
probability | Lim sup as a random variable in a bounded interval? | https://math.stackexchange.com/questions/2938028/lim-sup-as-a-random-variable-in-a-bounded-interval | <p>Let (Ω, <span class="math-container">$\mathcal{F}$</span>, <span class="math-container">$\mathbb{P}$</span>) be a probability space and, for each t ∈ [0, 1], let <span class="math-container">$X_t$</span> be a random variable on (Ω, F, P). For <span class="math-container">$\omega \in \Omega$</span></p>
<p>Y(<span cl... | <p>You want to check <span class="math-container">$Y$</span> is a measurable function, right?</p>
<p>There's some theorem saying if a class of functions <span class="math-container">$f_t(x)$</span> are measurable, then so does <span class="math-container">$\sup_t f_t(x)$</span>.
So if you consider <span class="math-co... | 381 |
probability | Standard deviation of two 8-sided dice rolled 10000 times | https://math.stackexchange.com/questions/2941691/standard-deviation-of-two-8-sided-dice-rolled-10000-times | <p>I am helping a friend with a study guide for a class, and one of the problems is asking about the theoretical mean and standard deviation. Two 8-sided dice with equal probabilities for 1, 2, 3, 4, 5, 6, 7, and 8, are rolled, and the sum of the two dice are recorded. </p>
<p>So I have a dataset that is the sum of th... | <p>You have assumed that each sum from <span class="math-container">$2$</span> through <span class="math-container">$16$</span> is equally probable, so each one shows up <span class="math-container">$\frac 1{15}$</span> of the time. That is not true. There are <span class="math-container">$64$</span> different rolls ... | 382 |
probability | n white and k black balls in m boxes probability of co-occurence | https://math.stackexchange.com/questions/2949935/n-white-and-k-black-balls-in-m-boxes-probability-of-co-occurence | <p>n white and k black balls are randomly and independently distributed amongst m boxes. There is no limit to the number of balls a box can contain.</p>
<p>As a result, there are four possible states for each box:</p>
<ol>
<li>Empty</li>
<li>black only</li>
<li>white only</li>
<li>black and white</li>
</ol>
<p>What ... | <p>It is important to be clear about the process for distributing balls between boxes. Just clearing this up can help you think about how to work out the probabilities.</p>
<p>Let's assume the procedure for assigning balls to boxes "randomly and independently" is this:</p>
<ul>
<li>Consider each of the <span class="m... | 383 |
probability | Let $\theta_n$ be a random variable that can be $\{\frac{1}{n},\frac{2}{n},...,\frac{n}{n}\}$ with equal probability $\frac{1}{n}$. | https://math.stackexchange.com/questions/2952037/let-theta-n-be-a-random-variable-that-can-be-frac1n-frac2n | <p>Let <span class="math-container">$\theta_n$</span> be a random variable that can be <span class="math-container">$\{\frac{1}{n},\frac{2}{n},...,\frac{n}{n}\}$</span> with equal probability <span class="math-container">$\frac{1}{n}$</span>. My question is where does <span class="math-container">$\theta_n$</span> conv... | <p>You can't restrict attention to the nice cases, you have to handle arbitrary <span class="math-container">$x \in \mathbb{R}$</span>. To do that, note that for <span class="math-container">$x \in [0,1]$</span>:</p>
<p><span class="math-container">$$P(\theta_n \leq x)=\frac{1}{n} |\{ k \in \mathbb{N} : k/n \leq x \}|... | 384 |
probability | What numbers in $[0,1]$ can be generated by tossing a fair coin? | https://math.stackexchange.com/questions/2953555/what-numbers-in-0-1-can-be-generated-by-tossing-a-fair-coin | <blockquote>
<p>What numbers in the interval <span class="math-container">$[0,1]$</span> can be generated by tossing a
fair coin? By generating a number using a coin, we mean finding an event that its probability is the given number.</p>
</blockquote>
<p>I think that any number in <span class="math-container">$[0,... | <p>Write <span class="math-container">$0.$</span> and now start tossing the coin. Write <span class="math-container">$1$</span> if it comes up heads and <span class="math-container">$0$</span> for tails. You will gradually spell out a binary representation of a number in the range <span class="math-container">$[0, 1]... | 385 |
probability | Transitive Conditional Probability Constraints | https://math.stackexchange.com/questions/2957640/transitive-conditional-probability-constraints | <p>Is there a nice rule describing how the values of <span class="math-container">$P(C|B)$</span> and <span class="math-container">$P(B|A)$</span> jointly constrain <span class="math-container">$P(C|A)$</span>? In particular, if I know that both <span class="math-container">$P(C|B)$</span> and <span class="math-contain... | <p>You cannot conclude anything about <span class="math-container">$P(C|A)$</span>, it can still be <span class="math-container">$0$</span>. For example, if
I am rolling a <span class="math-container">$6$</span> sided die and the events <span class="math-container">$A,B,C$</span> are</p>
<ul>
<li><span class="math-co... | 386 |
probability | How many turns it takes to cats in a chessboard jumps to the same square? | https://math.stackexchange.com/questions/2957812/how-many-turns-it-takes-to-cats-in-a-chessboard-jumps-to-the-same-square | <p>How can I solve this one? Ten cats are on the chessboard, every cat is on one of the square and the can be on the same square as well as in a different square. Every turn each cat jumps to the adjacent square with equal probability. They won't jump outside the board. So in the corner a cat can jump to three squares ... | <p>There are three different types of squares: </p>
<p>1) <span class="math-container">$6\times6=36$</span> central squares, each reachable from 8 different directions. Denote the probability of a cat taking a central square with <span class="math-container">$p_1$</span>.</p>
<p>2) <span class="math-container">$4\tim... | 387 |
probability | Probability that Amy gets more heads than Bob? | https://math.stackexchange.com/questions/2965489/probability-that-amy-gets-more-heads-than-bob | <p>Say that Amy tosses a coin 6 times, and Bob tosses a coin 5 times. What's the probability that Amy gets more heads than Bob does? </p>
| 388 | |
probability | What is the probability of obtaining a sum of $10$ in two consecutive throws of two dice in the first $8$ throws? | https://math.stackexchange.com/questions/2965715/what-is-the-probability-of-obtaining-a-sum-of-10-in-two-consecutive-throws-of | <p>We are playing with two dice until we get <span class="math-container">$10$</span> as the sum of the results two times in a row.</p>
<p>(i) What is the probability of having gotten this in <span class="math-container">$8$</span> throws?</p>
<p>(ii) What's the probability of having thrown a sum less than <span clas... | <p>(i) What is the probability of having gotten this in 8 throws?</p>
<p>Solution: You didn't consider the all possible events.
In order to calculate the required probability, you have to consider the throws before <span class="math-container">$7th$</span> for not getting the sum of <span class="math-container">$10$</... | 389 |
probability | If $\exists k>1$ odd such that $E(X-E(X))^k=0$, then is X symmetric? | https://math.stackexchange.com/questions/2967381/if-exists-k1-odd-such-that-ex-exk-0-then-is-x-symmetric | <p>If <span class="math-container">$\exists k>1$</span> odd such that <span class="math-container">$E(X-E(X))^k=0$</span>, then is X symmetric?</p>
<p>I know that the converse is true, in fact, if X is symmetric, then all odd moments will be zero.</p>
| <p>You can find a real-valued random variable <span class="math-container">$X$</span> that is not symmetric (I am guessing that it means <span class="math-container">$X-a$</span> and <span class="math-container">$a-X$</span> are identically distributed for some constant <span class="math-container">$a$</span>) and, for... | 390 |
probability | Find the probability that A drew it on the first draw | https://math.stackexchange.com/questions/2969428/find-the-probability-that-a-drew-it-on-the-first-draw | <p>A and B draw coins in turn without replacement from a bag containing <span class="math-container">$3$</span> dimes and <span class="math-container">$4$</span> nickels. A draws first. It is known that A drew the first dime. Find the probability that A drew it on the first draw.</p>
<p>I know that the probability of ... | <p>P<span class="math-container">$[$</span>A draws dime on first draw draws first dime<span class="math-container">$|$</span> A draws first dime<span class="math-container">$]=\dfrac{P(\mbox{A draws dime on the first draw })}{P(\mbox{A draws first dime})}$</span></p>
<p>So, <span class="math-container">$$P(\mbox{A dra... | 391 |
probability | Probability of arriving at a number after a certain number of tries with the following criteria | https://math.stackexchange.com/questions/2976252/probability-of-arriving-at-a-number-after-a-certain-number-of-tries-with-the-fol | <p>I'm not sure what field of math this is, I'm just interested in mathematics in daily life, here's a question that kept me thinking:</p>
<p>Lets say for a range between <span class="math-container">$1$</span> to <span class="math-container">$200$</span>, I randomly pick a number, for example I pick <span class="math... | <p>This is a nice question. Let <span class="math-container">$X=\{\text{number of picks needed to get to }200\}$</span>. First question is how many possible outcomes are there? </p>
<p>Well, if we play the game until we get to <span class="math-container">$200$</span>, then, we could either pick <span class="math-cont... | 392 |
probability | Normal Approximation to Poisson problem | https://math.stackexchange.com/questions/2982204/normal-approximation-to-poisson-problem | <p>Suppose we have code with <span class="math-container">$𝑛 = 100$</span> pages. The variable <span class="math-container">$𝑋𝑖$</span> is the number of errors on the page that is distributed Poisson meets the average of 1. Also, the number of errors per page is independent of the other page. Number The total errors... | <p>The key fact to use here is that the sum of independent Poisson RVs also has a Poisson distribution with a rate that is the sum of the individual rates. Specifically, since <span class="math-container">$X_i \sim Pois(1)$</span>, <span class="math-container">$Y = \sum_{i=1}^{100} X_i \sim Pois(100)$</span>. Since <sp... | 393 |
probability | How can I prove that $\mathbb P\{X\leq x\}=\int_{-\infty}^x f_X(t)dt$? | https://math.stackexchange.com/questions/2984419/how-can-i-prove-that-mathbb-p-x-leq-x-int-inftyx-f-xtdt | <p><span class="math-container">$$\mathbb P\{X\leq x\}=\int_{\Omega }\boldsymbol 1_{X^{-1}(-\infty,x]}(\omega )d\mathbb P(\omega )=\int_{\Omega }\boldsymbol 1_{(-\infty,x]}(X(\omega ))d\mathbb P(\omega ),$$</span></p>
<p>how can I continue ? I guess I have to do the substitution <span class="math-container">$t=X(\omeg... | 394 | |
probability | Problem to understand random variable : For example, $X(\omega )=1$ is really random ? | https://math.stackexchange.com/questions/2985937/problem-to-understand-random-variable-for-example-x-omega-1-is-really-ra | <p>I have difficulty to understand random variable. Let <span class="math-container">$(\Omega ,\mathcal F,\mathbb P)$</span> a probability space. Let say <span class="math-container">$\Omega =[0,1]$</span>, <span class="math-container">$\mathbb P=m$</span> the lebesgue measure and <span class="math-container">$\mathcal... | <p>Here's a perspective that was imparted to me by my thesis advisor: You have a function (i.e. <span class="math-container">$X$</span>, the random variable) on a domain (i.e. <span class="math-container">$\Omega$</span>, the sample space). It behaves just like all other functions you've ever encountered behave. The ca... | 395 |
probability | Could you calculate the odds of which would happen first? Sean Connery or an original cast member of the Simpsons dying? | https://math.stackexchange.com/questions/2991957/could-you-calculate-the-odds-of-which-would-happen-first-sean-connery-or-an-ori | <p>I was initially thinking about who would die first, Sean Connery or Roger Moore. Roger Moore passed on later so I got to thinking about could you calculate the odds of one instance versus one single group happening? In my example, what would happen first, an original cast member of the Simpsons dying or Sean Connery... | <p>I cannot address your specific example because I would have to know lots of details about the lives of Sean Connery and the cast members of The Simpsons to determine which people are in better/worse health and are thus more/less likely to die first, etc.</p>
<p>However, I will formalize, generalize, and answer your... | 396 |
probability | Find the density function for the time to failure $T$. | https://math.stackexchange.com/questions/2993505/find-the-density-function-for-the-time-to-failure-t | <blockquote>
<p>A machine has <span class="math-container">$5$</span> components and needs at least 3 working components
to function. Suppose that their lifetimes are independent
exponential(1). Find the density function for the time to failure <span class="math-container">$T$</span>.</p>
</blockquote>
<h3>attempt</h3>... | <p>No, <span class="math-container">$T$</span> cannot be binomial because it is a <strong>time</strong> to failure (of the system). If you want to <strong>count</strong> the number of components still operating at a given time, that at least is compatible with a discrete distribution with finite support.</p>
<p>If ea... | 397 |
probability | How to combine the probability of liking something with the one of it being liked? | https://math.stackexchange.com/questions/2995012/how-to-combine-the-probability-of-liking-something-with-the-one-of-it-being-like | <p>How to combine the probability of liking something with the one of it being liked?</p>
<p>I'd like to estimate the probability of a person liking a dish by only having the following two bits of information given:</p>
<ul>
<li>The person likes 70% (<code>like_rate = 0.7</code>) of all dishes you offer him.</li>
<li... | <p>At the moment the question is a bit under defined as there is a lot of subjectivity in how one is to interpret it. For instance </p>
<ol>
<li><p>Is the person who likes 70% of dishes from the same population as the people who have already tried it? I.e. is it the case that in general any member of the population li... | 398 |
probability | Find $P(X=Y+1)$ | https://math.stackexchange.com/questions/2997775/find-px-y1 | <p>Suppose that X and Y are integer valued random variables with joint probability mass function given by
<span class="math-container">$p_{X,Y}(a, b)=\begin{cases} \frac{1}{4a}, & 1\leq b\leq a\leq 4\\ 0, & \text{otherwise}. \end{cases}.$</span></p>
<p>(b) Find the marginal probability mass functions of X and... | <p>The table is not hard. Each cell in 4 rows (<span class="math-container">$1\leq a\leq 4$</span>) of <span class="math-container">$a$</span> columns (<span class="math-container">$1\leq b\leq a$</span>), contains <span class="math-container">$\tfrac 1{4a}$</span>, the rest of the cells are zero .</p>
<p><spa... | 399 |
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