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# A First Course in PROBABILITY
## NINTH EDITION
# SHELDON ROSS
# A First Course in Probability
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# A First Course in Probability
Ninth Edition
## Sheldon Ross <br> University of Southern California
Boston Columbus Indianapolis New York San Francisco
Upper Saddle River Amsterdam... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 1 |
otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 021... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 2 |
Introduction I
I. 2 The Basic Principle of Counting 2
I. 3 Permutations 3
I. 4 Combinations 5
I. 5 Multinomial Coefficients 9
I. 6 The Number of Integer Solutions of Equations 12
Summary 15
Problems 15
Theoretical Exercises 17
Self-Test Problems and Exercises 19
2 Axioms of Probability 21
2.I Introduction 21
2.2 Sample... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 3 |
Problems and Exercises 109
4 RANDOM VARIABLES 112
4.I Random Variables 112
4.2 Discrete Random Variables 116
4.3 Expected Value 119
4.4 Expectation of a Function of a Random Variable 121
4.5 Variance 125
4.6 The Bernoulli and Binomial Random Variables 127
4.7 The Poisson Random Variable 135
4.8 Other Discrete Probabil... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 4 |
Theoretical Exercises 214
Self-Test Problems and Exercises 217
6 Jointly Distributed Random
VARIABLES 220
6.1 Joint Distribution Functions..... 220
6.2 Independent Random Variables..... 228
6.3 Sums of Independent Random Variables..... 239
6.4 Conditional Distributions: Discrete Case..... 248
6.5 Conditional Distributi... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 5 |
Normal Random Variables..... 345
7.9 General Definition of Expectation..... 349
Summary..... 351
Problems..... 352
Theoretical Exercises..... 359
Self-Test Problems and Exercises..... 363
8 Limit Theorems 367
8.1 Introduction..... 367
8.2 Chebyshev's Inequality and the Weak Law of Large Numbers..... 367
8.3 The Central... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 6 |
Test Problems and Exercises..... 413
10 Simulation..... 415
10.1 Introduction..... 415
10.2 General Techniques for Simulating Continuous Random Variables..... 417
10.3 Simulating from Discrete Distributions..... 424
10.4 Variance Reduction Techniques..... 426
Summary..... 430
Problems..... 430
Self-Test Problems and Ex... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 7 |
, it is nevertheless true that probability theory has become a tool of fundamental importance to nearly all scientists, engineers, medical practitioners, jurists, and industrialists. In fact, the enlightened individual had learned to ask not "Is it so?" but rather "What is the probability that it is so?"
# General App... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 8 |
and 6. Discrete random variables are dealt with in Chapter 4, continuous random variables in Chapter 5, and jointly distributed random variables in Chapter 6. The important concepts of the expected value and the variance of a random variable are introduced in Chapters 4 and 5, and these quantities are then determined f... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 9 |
is not too large, then the answer is yes.
Figure 3.3
The argument runs as follows: Suppose that each edge is, independently, equally likely to be colored either red or blue. That is, each edge is red with probability $\frac{1}{2}$. Number the $\binom{n}{k}$ sets of $k$ vertices and define the events $E_{i}, i=1, \ldo... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 10 |
i} E_{i}\right)$, the probability that there is a set of $k$ vertices all of whose connecting edges are similarly colored, satisfies
$$
P\left(\bigcup_{i} E_{i}\right) \leq\binom{ n}{k}\left(\frac{1}{2}\right)^{k(k-1) / 2-1}
$$
Hence, if
$$
\binom{n}{k}\left(\frac{1}{2}\right)^{k(k-1) / 2-1}<1
$$
or, equivalently, i... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 11 |
no information about how to obtain such a scheme (although one possibility would be simply to choose the colors at random, check to see if the resulting coloring satisfies the property, and repeat the procedure until it does).
(b) The method of introducing probability into a problem whose statement is purely determinis... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 12 |
\leq 1$. The left-side inequality is obvious, whereas the right side follows because $E F \subset F$, which implies that $P(E F) \leq P(F)$. Part (b) follows because
$$
P(S \mid F)=\frac{P(S F)}{P(F)}=\frac{P(F)}{P(F)}=1
$$
[^0]
[^0]: ${ }^{\dagger}$ See N. Alon, J. Spencer, and P. Erdos, The Probabilistic Method ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 13 |
{\infty} P\left(E_{i} F\right)}{P(F)} \\
& =\sum_{1}^{\infty} P\left(E_{i} \mid F\right)
\end{aligned}
$$
where the next-to-last equality follows because $E_{i} E_{j}=\varnothing$ implies that $E_{i} F E_{j} F=\emptyset$.
If we define $Q(E)=P(E \mid F)$, then, from Proposition 5.1, $Q(E)$ may be regarded as a probabi... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 14 |
by $Q\left(E_{1} \mid E_{2}\right)=Q\left(E_{1} E_{2}\right) /$ $Q\left(E_{2}\right)$, then, from Equation (3.1), we have
$$
Q\left(E_{1}\right)=Q\left(E_{1} \mid E_{2}\right) Q\left(E_{2}\right)+Q\left(E_{1} \mid E_{2}^{\mathrm{c}}\right) Q\left(E_{2}^{\mathrm{c}}\right)
$$
Since
$$
\begin{aligned}
Q\left(E_{1} \mi... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 15 |
to
$$
P\left(E_{1} \mid F\right)=P\left(E_{1} \mid E_{2} F\right) P\left(E_{2} \mid F\right)+P\left(E_{1} \mid E_{2}^{\mathrm{c}} F\right) P\left(E_{2}^{\mathrm{c}} \mid F\right)
$$
Example 5a
Consider Example 3a, which is concerned with an insurance company that believes that people can be divided into two distinct... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 16 |
\left(A_{2} \mid A_{1}\right)=P\left(A_{2} \mid A A_{1}\right) P\left(A \mid A_{1}\right)+P\left(A_{2} \mid A^{c} A_{1}\right) P\left(A^{c} \mid A_{1}\right)
$$
Now,
$$
P\left(A \mid A_{1}\right)=\frac{P\left(A_{1} A\right)}{P\left(A_{1}\right)}=\frac{P\left(A_{1} \mid A\right) P(A)}{P\left(A_{1}\right)}
$$
However,... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 17 |
left(A_{2} \mid A^{c} A_{1}\right)=.2$, it follows that
$$
P\left(A_{2} \mid A_{1}\right)=(.4) \frac{6}{13}+(.2) \frac{7}{13} \approx.29
$$
A female chimp has given birth. It is not certain, however, which of two male chimps is the father. Before any genetic analysis has been performed, it is believed that the probab... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 18 |
event that male number $i, i=1,2$, is the father, and let $B_{A, a}$ be the event that the baby chimp has the gene pair $(A, a)$. Then, $P\left(M_{1} \mid B_{A, a}\right)$ is obtained as follows:
$$
\begin{aligned}
P\left(M_{1} \mid B_{A, a}\right) & =\frac{P\left(M_{1} B_{A, a}\right)}{P\left(B_{A, a}\right)} \\
& =\... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 19 |
intuitive
because it is more likely that the baby would have gene pair $(A, a)$ if $M_{1}$ is true than if $M_{2}$ is true (the respective conditional probabilities being 1 and $1 / 2$ ).
The next example deals with a problem in the theory of runs.
## Example
5c
Independent trials, each resulting in a success with ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 20 |
successes, we obtain
$$
P(E \mid H)=P(E \mid F H) P(F \mid H)+P\left(E \mid F^{c} H\right) P\left(F^{c} \mid H\right)
$$
On the one hand, clearly, $P(E \mid F H)=1$; on the other hand, if the event $F^{c} H$ occurs, then the first trial would result in a success, but there would be a failure some time during the next... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 21 |
2 through $m$ are all failures. Then,
$$
P\left(E \mid H^{c}\right)=P\left(E \mid G H^{c}\right) P\left(G \mid H^{c}\right)+P\left(E \mid G^{c} H^{c}\right) P\left(G^{c} \mid H^{c}\right)
$$
Now, $G H^{c}$ is the event that the first $m$ trials all result in failures, so $P\left(E \mid G H^{c}\right)=0$. Also, if $G^... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 22 |
$$
P(E \mid H)=\frac{p^{n-1}}{p^{n-1}+q^{m-1}-p^{n-1} q^{m-1}}
$$
and
$$
P\left(E \mid H^{c}\right)=\frac{\left(1-q^{m-1}\right) p^{n-1}}{p^{n-1}+q^{m-1}-p^{n-1} q^{m-1}}
$$
Thus,
$$
\begin{aligned}
P(E) & =p P(E \mid H)+q P\left(E \mid H^{c}\right) \\
& =\frac{p^{n}+q p^{n-1}\left(1-q^{m-1}\right)}{p^{n-1}+q^{m-1}-... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 23 |
q$ interchanged and $n$ and $m$ interchanged. Hence, this probability would equal
$$
\begin{aligned}
& P\left\{\text { run of } m \text { failures before a run of } n \text { successes }\right\} \\
& =\frac{q^{m-1}\left(1-p^{n}\right)}{q^{m-1}+p^{n-1}-q^{m-1} p^{n-1}}
\end{aligned}
$$
Since Equations (5.7) and (5.8) ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 24 |
(a) no matches?
(b) exactly $k$ matches?
Solution (a) Let $E$ denote the event that no matches occur, and to make explicit the dependence on $n$, write $P_{n}=P(E)$. We start by conditioning on whether or not the first man selects his own hat-call these events $M$ and $M^{c}$, respectively. Then,
$$
P_{n}=P(E)=P(E \m... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 25 |
the extra man. Because the second event has probability $[1 /(n-1)] P_{n-2}$, we have
$$
P\left(E \mid M^{c}\right)=P_{n-1}+\frac{1}{n-1} P_{n-2}
$$
Thus, from Equation (5.9),
$$
P_{n}=\frac{n-1}{n} P_{n-1}+\frac{1}{n} P_{n-2}
$$
or, equivalently,
$$
P_{n}-P_{n-1}=-\frac{1}{n}\left(P_{n-1}-P_{n-2}\right)
$$
Howeve... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 26 |
3}-P_{2}\right)}{4}=\frac{1}{4!} \quad \text { or } \quad P_{4}=\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}
\end{aligned}
$$
and, in general,
$$
P_{n}=\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\cdots+\frac{(-1)^{n}}{n!}
$$
(b) To obtain the probability of exactly $k$ matches, we consider any fixed group of $k$ men. The prob... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 27 |
!}+\cdots+\frac{(-1)^{n-k}}{(n-k)!}}{k!}
$$
An important concept in probability theory is that of the conditional independence of events. We say that the events $E_{1}$ and $E_{2}$ are conditionally independent given $F$ if given that $F$ occurs, the conditional probability that $E_{1}$ occurs is unchanged by informat... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 28 |
was accident prone. The following
example, sometimes referred to as Laplace's rule of succession, further illustrates the concept of conditional independence.
## Example Laplace's rule of succession
There are $k+1$ coins in a box. When flipped, the $i$ th coin will turn up heads with probability $i / k, i=0,1, \ldots... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 29 |
the outcomes will be conditionally independent, with each one resulting in a head with probability $i / k$. Hence,
$$
P\left(H \mid F_{n} C_{i}\right)=P\left(H \mid C_{i}\right)=\frac{i}{k}
$$
Also,
$$
P\left(C_{i} \mid F_{n}\right)=\frac{P\left(C_{i} F_{n}\right)}{P\left(F_{n}\right)}=\frac{P\left(F_{n} \mid C_{i}\... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 30 |
i=0}^{k}\left(\frac{i}{k}\right)^{n+1} & \approx \int_{0}^{1} x^{n+1} d x=\frac{1}{n+2} \\
\frac{1}{k} \sum_{j=0}^{k}\left(\frac{j}{k}\right)^{n} & \approx \int_{0}^{1} x^{n} d x=\frac{1}{n+1}
\end{aligned}
$$
So, for $k$ large,
$$
P\left(H \mid F_{n}\right) \approx \frac{n+1}{n+2}
$$
## Example Updating information ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 31 |
_{j} P\left(E \mid H_{j}\right) P\left(H_{j}\right)}
$$
Suppose now that we learn first that $E_{1}$ has occurred and then that $E_{2}$ has occurred. Then, given only the first piece of information, the conditional probability that $H_{i}$ is the true hypothesis is
$$
P\left(H_{i} \mid E_{1}\right)=\frac{P\left(E_{1}... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 32 |
1} E_{2} \mid H_{j}\right) P\left(H_{j}\right)}
$$
One might wonder, however, when one can compute $P\left(H_{i} \mid E_{1} E_{2}\right)$ by using the right side of Equation (5.13) with $E=E_{2}$ and with $P\left(H_{j}\right)$ replaced by $P\left(H_{j} \mid E_{1}\right)$, $j=1, \ldots, n$. That is, when is it legitima... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 33 |
aligned}
P\left(H_{i} \mid E_{1} E_{2}\right) & =\frac{P\left(E_{2} \mid H_{i}\right) P\left(E_{1} \mid H_{i}\right) P\left(H_{i}\right)}{P\left(E_{1} E_{2}\right)} \\
& =\frac{P\left(E_{2} \mid H_{i}\right) P\left(E_{1} H_{i}\right)}{P\left(E_{1} E_{2}\right)} \\
& =\frac{P\left(E_{2} \mid H_{i}\right) P\left(H_{i} \m... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 34 |
n} P\left(H_{i} \mid E_{1} E_{2}\right)=\sum_{i=1}^{n} \frac{P\left(E_{2} \mid H_{i}\right) P\left(H_{i} \mid E_{1}\right)}{Q(1,2)}
$$
showing that
$$
Q(1,2)=\sum_{i=1}^{n} P\left(E_{2} \mid H_{i}\right) P\left(H_{i} \mid E_{1}\right)
$$
and yielding the result
$$
P\left(H_{i} \mid E_{1} E_{2}\right)=\frac{P\left(E_... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 35 |
that to sequentially update the probability that coin 1 is the one being flipped, given the results of the previous flips, all that must be saved after each new flip is the conditional probability that coin 1 is the coin being used. That is, it is not necessary to keep track of all earlier results.
# Summary
For eve... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 36 |
The identity
$$
\frac{P(H \mid E)}{P\left(H^{c} \mid E\right)}=\frac{P(H) P(E \mid H)}{P\left(H^{c}\right) P\left(E \mid H^{c}\right)}
$$
shows that when new evidence $E$ is obtained, the value of the odds of $H$ becomes its old value multiplied by the ratio of the conditional probability of the new evidence when $H$... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 37 |
P\left(E \mid F_{i}\right) P\left(F_{i}\right)
$$
which is called the law of total probability.
If $P(E F)=P(E) P(F)$, then we say that the events $E$ and $F$ are independent. This condition is equivalent to $P(E \mid F)=P(E)$ and to $P(F \mid E)=P(F)$. Thus, the events $E$ and $F$ are independent if knowledge of the ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 38 |
probability that the first one lands on 6 given that the sum of the dice is $i$? Compute for all values of $i$ between 2 and 12.
3.3. Use Equation (2.1) to compute in a hand of bridge the conditional probability that East has 3 spades given that North and South have a combined total of 8 spades.
3.4. What is the probab... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 39 |
also presents such probability inequalities as Markov's inequality, Chebyshev's inequality, and Chernoff bounds. The final section of Chapter 8 gives a bound on the error involved when a probability concerning a sum of independent Bernoulli random variables is approximated by the corresponding probability of a Poisson ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 40 |
the time to contact me with comments for improving the text: Amir Ardestani, Polytechnic University of Teheran; Joe Blitzstein, Harvard University; Peter Nuesch, University of Lausaunne; Joseph Mitchell, SUNY, Stony Brook; Alan Chambless, actuary; Robert Kriner; Israel David, Ben-Gurion University; T. Lim, George Mason... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 41 |
University Richard Bass, University of Connecticut Robert Bauer, University of Illinois at Urbana-Champaign Phillip Beckwith, Michigan Tech Arthur Benjamin, Harvey Mudd College Geoffrey Berresford, Long Island University Baidurya Bhattacharya, University of Delaware Howard Bird, St. Cloud State University Shahar Boneh,... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 42 |
of California, Irvine
Chuanshu Ji, University of North Carolina, Chapel Hill
Robert Keener, University of Michigan
*Richard Laugesen, University of Illinois
Fred Leysieffer, Florida State University
Thomas Liggett, University of California, Los Angeles
Helmut Mayer, University of Georgia
Bill McCormick, University of G... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 43 |
1.3 Permutations
1.4 Combinations
1.5 Multinomial Coefficients
1.6 The Number of Integer Solutions of Equations
## I.I Introduction
Here is a typical problem of interest involving probability: A communication system is to consist of $n$ seemingly identical antennas that are to be lined up in a linear order. The resul... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 44 |
6}=\frac{1}{2}$ as the desired probability. In the case of general $n$ and $m$, we could compute the probability that the system is functional in a similar fashion. That is, we could count the number of configurations that result in the system's being functional and then divide by the total number of all possible confi... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 45 |
all the possible outcomes of the two experiments; that is,
$$
\begin{aligned}
& (1,1),(1,2), \ldots,(1, n) \\
& (2,1),(2,2), \ldots,(2, n) \\
& \vdots \\
& (m, 1),(m, 2), \ldots,(m, n)
\end{aligned}
$$
where we say that the outcome is $(i, j)$ if experiment 1 results in its $i$ th possible outcome and experiment 2 th... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 46 |
first one may result in any of $n_{1}$ possible outcomes; and if, for each of these $n_{1}$ possible outcomes, there are $n_{2}$ possible outcomes of the second experiment; and if, for each of the possible outcomes of the first two experiments, there are $n_{3}$ possible outcomes of the third experiment; and if $\ldots... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 47 |
| Example <br> 2e | Solution By the generalized version of the basic principle, the answer is $26 \cdot 26$. $26 \cdot 10 \cdot 10 \cdot 10 \cdot 10=175,760,000$. |
| Example <br> 2d | How many functions defined on $n$ points are possible if each functional value is either 0 or 1? |
| | Solution Let the points be $1,2... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 48 |
b, b a c, b c a, c a b$, and $c b a$. Each arrangement is known as a permutation. Thus, there are 6 possible permutations of a set of 3 objects. This result could also have been obtained from the basic principle, since the first object in the permutation can be any of the 3, the second object in the permutation can the... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 49 |
4 women. An examination is given, and the students are ranked according to their performance. Assume that no two students obtain the same score.
(a) How many different rankings are possible?
(b) If the men are ranked just among themselves and the women just among themselves, how many different rankings are possible?
S... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 50 |
each possible ordering of the subjects, there are 4! 3! 2! 1! possible arrangements. Hence, as there are 4! possible orderings of the subjects, the desired answer is 4! 4! 3! 2! $1!=6912$.
We shall now determine the number of permutations of a set of $n$ objects when certain of the objects are indistinguishable from o... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 51 |
_{1} P_{2} E_{2} P_{3} E_{1} R \\
P_{1} P_{3} E_{1} P_{2} E_{2} R & P_{1} P_{3} E_{2} P_{2} E_{1} R \\
P_{2} P_{1} E_{1} P_{3} E_{2} R & P_{2} P_{1} E_{2} P_{3} E_{1} R \\
P_{2} P_{3} E_{1} P_{1} E_{2} R & P_{2} P_{3} E_{2} P_{1} E_{1} R \\
P_{3} P_{1} E_{1} P_{2} E_{2} R & P_{3} P_{1} E_{2} P_{2} E_{1} R \\
P_{3} P_{2... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 52 |
in Example 3d shows that there are
$$
\frac{n!}{n_{1}!n_{2}!\cdots n_{r}!}
$$
different permutations of $n$ objects, of which $n_{1}$ are alike, $n_{2}$ are alike, $\ldots, n_{r}$ are alike.
A chess tournament has 10 competitors, of which 4 are Russian, 3 are from the United States, 2 are from Great Britain, and 1 i... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 53 |
3 could be selected from the 5 items $A, B, C, D$, and $E$? To answer this question, reason as follows: Since there are 5 ways to select the initial item, 4 ways to then select the next item, and 3 ways to select the final item, there are thus $5 \cdot 4 \cdot 3$ ways of selecting the group of 3 when the order in which... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 54 |
r$ items that could be formed from a set of $n$ items is
$$
\frac{n(n-1) \cdots(n-r+1)}{r!}=\frac{n!}{(n-r)!r!}
$$
## Notation and terminology
We define $\binom{n}{r}$, for $r \leq n$, by
$$
\binom{n}{r}=\frac{n!}{(n-r)!r!}
$$
and say that $\binom{n}{r}$ (read as " $n$ choose $r$ ") represents the number of possib... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 55 |
Using that $0!=1$, note that $\binom{n}{n}=\binom{n}{0}=\frac{n!}{0!n!}=1$, which is consistent with the preceding interpretation because in a set of size $n$ there is exactly 1 subset of size $n$ (namely, the entire set), and exactly one subset of size 0 (namely the empty set). A useful convention is to define $\binom... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 56 |
}{3}=$ $\binom{5 \cdot 4}{2 \cdot 1} \frac{7 \cdot 6 \cdot 5}{3 \cdot 2 \cdot 1}=350$ possible committees consisting of 2 women and 3 men.
Now suppose that 2 of the men refuse to serve together. Because a total of $\binom{2}{2}\binom{5}{1}=5$ out of the $\binom{7}{3}=35$ possible groups of 3 men contain both of the fe... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 57 |
in the $n-m+1$ possible positions-represented in Figure 1.1 by carets-between the $n-m$ functional antennas, we must select $m$ of these in which to put the defective antennas. Hence, there are $\binom{n-m+1}{m}$ possible orderings in which there is at least one functional antenna between any two defective ones.
Figu... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 58 |
often referred to as binomial coefficients because of their prominence in the binomial theorem.
# The binomial theorem
$$
(x+y)^{n}=\sum_{k=0}^{n}\binom{n}{k} x^{k} y^{n-k}
$$
We shall present two proofs of the binomial theorem. The first is a proof by mathematical induction, and the second is a proof based on comb... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 59 |
-k}
\end{aligned}
$$3.5. An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without replacement, what is the probability that the first 2 selected are white and the last 2 black?
3.6. Consider an urn containing 12 balls, of which 8 are white. A sample of size 4 is to be drawn with replace... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 60 |
card selected is a spade given that the second and third cards are spades.
3.11. Two cards are randomly chosen without replacement from an ordinary deck of 52 cards. Let $B$ be the event that both cards are aces, let $A_{s}$ be the event that the ace of spades is chosen, and let $A$ be the event that at least one ace i... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 61 |
13. Suppose that an ordinary deck of 52 cards (which contains 4 aces) is randomly divided into 4 hands of 13 cards each. We are interested in determining $p$, the probability that each hand has an ace. Let $E_{i}$ be the event that the $i$ th hand has exactly one ace. Determine $p=$ $P\left(E_{1} E_{2} E_{3} E_{4}\righ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 62 |
. If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby survives?
3.17. In a certain community, 36 percent of the families own a dog and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat. What is
(a) the probability... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 63 |
the end of a year. If 62 percent of the original class was male,
(a) what percentage of those attending the party were women?
(b) what percentage of the original class attended the party?
3.20. Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 64 |
couples is randomly chosen, what is
(a) the probability that the husband earns less than $\$ 25,000$?
(b) the conditional probability that the wife earns more than $\$ 25,000$ given that the husband earns more than this amount?
(c) the conditional probability that the wife earns more than $\$ 25,000$ given that the hus... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 65 |
is then randomly selected from urn II. What is
(a) the probability that the ball selected from urn II is white?
(b) the conditional probability that the transferred ball was white given that a white ball is selected from urn II?
3.24. Each of 2 balls is painted either black or gold and then placed in an urn. Suppose th... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 66 |
a person under the age of 50 spends in the streets, and let $\alpha_{2}$ be the corresponding value for those over 50. What quantity does the method suggested estimate? When is the estimate approximately equal to $p$?
3.26. Suppose that 5 percent of men and 0.25 percent of women are color blind. A color-blind person is... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 67 |
the card following it is the
(a) ace of spades?
(b) two of clubs?
3.29. There are 15 tennis balls in a box, of which 9 have not previously been used. Three of the balls are randomly chosen, played with, and then returned to the box. Later, another 3 balls are randomly chosen from the box. Find the probability that none... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 68 |
, the doctor is not to call. In this way, even if the doctor doesn't call, the news is not necessarily bad. Let $\alpha$ be the probability that the tumor is cancerous; let $\beta$ be the conditional probability that the tumor is cancerous given that the doctor does not call.
(a) Which should be larger, $\alpha$ or $\b... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 69 |
that the new evidence is subject to different possible interpretations and in fact shows only that it is 90 percent likely that the criminal possesses the characteristic in question. In this case, how likely would it be that the suspect is guilty (assuming, as before, that he has the characteristic)?
3.35. With probabi... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 70 |
same coin a second time and, again, it shows heads. Now what is the probability that it is the fair coin?
(c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?
3.38. Urn $A$ has 5 white and 7 black balls. Urn $B$ has 3 white and 12 black balls. We... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 71 |
shuffled and then divided into two halves of 26 cards each. A card is drawn from one of the halves; it turns out to be an ace. The ace is then placed in the second half-deck. The half is then shuffled, and a card is drawn from it. Compute the probability that this drawn card is an ace.
Hint: Condition on whether or not... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 72 |
.44. Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner $A$ asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 73 |
male is $\alpha, 0<\alpha<1$. A policyholder is randomly chosen. If $A_{i}$ denotes the event that this policyholder will make a claim in year $i$, show that
$$
P\left(A_{2} \mid A_{1}\right)>P\left(A_{1}\right)
$$
Give an intuitive explanation of why the preceding inequality is true.
3.47. An urn contains 5 white an... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 74 |
are indicative of cancer, the test is notoriously unreliable. Indeed, the probability that a noncancerous man will have an elevated PSA level is approximately.135, increasing to
approximately.268 if the man does have cancer. If, on the basis of other factors, a physician is 70 percent certain that a male has prostate c... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 75 |
a letter of recommendation for a new job. She estimates that there is an 80 percent chance that she will get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation. She further estimates that th... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 76 |
25 |.10 |
| Thursday |.15 |.15 |
| Friday |.10 |.20 |
She estimates that her probability of being accepted is.6.
(a) What is the probability that she receives mail on Monday?
(b) What is the conditional probability that she receives mail on Tuesday given that she does not receive mail on Monday?
(c) If there is no mail... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 77 |
telephone book.
(c) $E$ is the event that a man is under 6 feet tall, and $F$ is the event that he weighs more than 200 pounds.
(d) $E$ is the event that a woman lives in the United States, and $F$ is the event that she lives in the Western Hemisphere.
(e) $E$ is the event that it will rain tomorrow, and $F$ is the eve... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 78 |
assumed to be independent.
(a) What is the probability that after 2 days the stock will be at its original price?
(b) What is the probability that after 3 days the stock's price will have increased by 1 unit?
(c) Given that after 3 days the stock's price has increased by 1 unit, what is the probability that it went up ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 79 |
, H$?
(c) What is the probability that the pattern $T, H, H, H$ occurs before the pattern $H, H, H, H$?
Hint for part (c): How can the pattern $H, H, H, H$ occur first?
3.60. The color of a person's eyes is determined by a single pair of genes. If they are both blue-eyed genes, then the person will have blue eyes; if t... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 80 |
$. Only those people who receive the $a$ gene from both parents will be albino. Persons having the gene pair $A, a$ are normal in appearance and, because they can pass on the trait to their offspring, are called carriers. Suppose that a normal couple has two children, exactly one of whom is an albino. Suppose that the ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 81 |
the results of the shots are independent and that each shot of $A$ will hit $B$ with probability $p_{A}$, and each shot of $B$ will hit $A$ with probability $p_{B}$. What is
(a) the probability that $A$ is not hit?
(b) the probability that both duelists are hit?
(c) the probability that the duel ends after the $n$th ro... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 82 |
.66. The probability of the closing of the $i$ th relay in the circuits shown in Figure 3.4 is given by $p_{i}, i=1,2,3,4,5$. If all relays function independently, what is the probability that a current flows between $A$ and $B$ for the respective circuits?
Hint for (b): Condition on whether relay 3 closes.
3.67. An e... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 83 |
, find the conditional probability that relays 1 and 2 are both closed given that a current flows from $A$ to $B$.
3.69. A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercas... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 84 |
the contributions of the organism's mate. In a mating between organisms having genotypes $a A, b B, c C$, $d D, e E$ and $a a, b B, c c, D d$, ee what is the probability that the progeny will (i) phenotypically and (ii) genotypically resemble
(a) the first parent?
(b) the second parent?
(c) either parent?
(d) neither p... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 85 |
with the Dodgers, and the 3 remaining games of the Braves were against the San Diego Padres. Suppose that the outcomes of all remaining games are independent
and each game is equally likely to be won by either participant. For each team, what is the probability that it will win the division title? If two teams tie for ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 86 |
sex.
(b) The 3 eldest are boys and the others girls.
(c) Exactly 3 are boys.
(d) The 2 oldest are girls.
(e) There is at least 1 girl.
3.74. $A$ and $B$ alternate rolling a pair of dice, stopping either when $A$ rolls the sum 9 or when $B$ rolls the sum 6. Assuming that $A$ rolls first, find the probability that the fi... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 87 |
77. Consider an unending sequence of independent trials, where each trial is equally likely to result in any of the outcomes 1,2, or 3. Given that outcome 3 is the last of the
three outcomes to occur, find the conditional probability that
(a) the first trial results in outcome 1 ;
(b) the first two trials both result i... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 88 |
, and so on, until a single winner remains. Consider two specified contestants, $A$ and $B$, and define the events $A_{i}, i \leq n, E$ by
$$
\begin{aligned}
A_{i}: & A \text { plays in exactly } i \text { contests } \\
E: & A \text { and } B \text { never play each other }
\end{aligned}
$$
(a) Find $P\left(A_{i}\rig... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 89 |
$$
For another approach to solving this problem, note that there are a total of $2^{n}-1$ games played.
(d) Explain why $2^{n}-1$ games are played.
Number these games, and let $B_{i}$ denote the event that $A$ and $B$ play each other in game $i, i=1, \ldots, 2^{n}-1$.
(e) What is $P\left(B_{i}\right)$?
(f) Use part (... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 90 |
is the first one to get
(a) 2 heads in a row;
(b) a total of 2 heads;
(c) 3 heads in a row;
(d) a total of 3 heads.
In each case, find the probability that $A$ wins.
3.83. Die $A$ has 4 red and 2 white faces, whereas die $B$ has 2 red and 4 white faces. A fair coin is flipped once. If it lands on heads, the game conti... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 91 |
not replaced.
3.85. Repeat Problem 3.84 when each of the 3 players selects from his own urn. That is, suppose that there are 3 different urns of 12 balls with 4 white balls in each urn.
3.86. Let $S=\{1,2, \ldots, n\}$ and suppose that $A$ and $B$ are, independently, equally likely to be any of the $2^{n}$ subsets (inc... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 92 |
events that the key is in the
right-hand pocket of the jacket and that it is in the lefthand pocket. Also, let $S_{R}$ be the event that a search of the right-hand jacket pocket will be successful in finding the key, and let $U_{L}$ be the event that a search of the lefthand jacket pocket will be unsuccessful and, thus... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 93 |
}\right) \\
& +P\left(S_{R} \mid R^{c} U_{L}\right) P\left(R^{c} \mid U_{L}\right)
\end{aligned}
$$
3.88. In Example 5e, what is the conditional probability that the $i$ th coin was selected given that the first $n$ trials all result in heads?
3.89. In Laplace's rule of succession (Example 5e), are the outcomes of the... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 94 |
, each of which results in any of the outcomes 0,1, or 2, with respective probabilities $p_{0}, p_{1}$, and $p_{2}, \sum_{i=0}^{2} p_{i}=1$, are performed. Find the probability that outcomes 1 and 2 both occur at least once.
# Theoretical Exercises
3.1. Show that if $P(A)>0$, then
$$
P(A B \mid A) \geq P(A B \mid A... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 95 |
is more likely than method 2 to result in the choice of a firstborn child.
Hint: In solving this problem, you will need to show that
$$
\sum_{i=1}^{k} i n_{i} \sum_{j=1}^{k} \frac{n_{j}}{j} \geq \sum_{i=1}^{k} n_{i} \sum_{j=1}^{k} n_{j}
$$
To do so, multiply the sums and show that for all pairs $i, j$, the coefficien... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 96 |
_{i}} & \text { if } j=i
\end{array}
$$
3.5. (a) Prove that if $E$ and $F$ are mutually exclusive, then
$$
P(E \mid E \cup F)=\frac{P(E)}{P(E)+P(F)}
$$
(b) Prove that if $E_{i}, i \geq 1$ are mutually exclusive, then
$$
P\left(E j \mid \cup_{i=1}^{\infty} E_{i}\right)=\frac{P\left(E_{j}\right)}{\sum_{i=1}^{\infty} ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 97 |
the experiment continues until all the balls are removed, and consider the last ball withdrawn.
(b) A pond contains 3 distinct species of fish, which we will call the Red, Blue, and Green fish. There are $r$ Red, $b$ Blue, and $g$ Green fish. Suppose that the fish are removed from the pond in a random order. (That is, ... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 98 |
{ and } \quad P(B \mid C)>P\left(B \mid C^{c}\right)
$$
either prove that $P(A B \mid C)>P\left(A B \mid C^{c}\right)$ or give a counterexample by defining events $A, B$, and $C$ for which that relationship is not true.
Hint: Let $C$ be the event that the sum of a pair of dice is 10 ; let $A$ be the event that the fir... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 99 |
Given that a woman
has a positive mammography, what is the probability she has breast cancer?
3.11. In each of $n$ independent tosses of a coin, the coin lands on heads with probability $p$. How large need $n$ be so that the probability of obtaining at least one head is at least $\frac{1}{2}$?
3.12. Show that $0 \leq a... | A First Course in Probability (Sheldon Ross) (Z-Library) (converted to markdown via Mistral OCR) - chunk 100 |
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