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Sometimes we make guesses because we wish, with our limited knowledge,
to say as much as we cøn about some situation. Really, any generalization is
in the nature of a guess. Any physical theory is a kind of guesswork. There
are good guesses and there are bad guesses. “The theory of probability is a
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system for making better guesses. The language of probability allows us to speak
quantitatively about some situation which may be highly variable, but which
does have some consistent average behavior.
Let us consider the flipping of a coïn. If the toss—and the coin——are “honest,”
we have no way of knowing what to expect for the outcome of any particular toss.
Yet we would feel that in a large number of tosses there should be about equal
numbers oŸ heads and tails. We say: “The probability that a toss will land heads
is 0.5.”
W© speak of probability only for observations that we contemplate being made
in the future. Pựụ the “probabtlitU” oƒƑ a parlicular outcome oƒ an obser0atlion tue
mean our' estimate for the most likelU [raclion oƒ a tuumber öƒ repeated obseruations
that tuiil uicld that particular ou‡come. TÝ we imagine repeating an observatlon——
such as looking at a freshly tossed coin——/ times, and ïf we call WA our estimafe
of the most likely number of our observations that will give some specified result A,
say the result “heads,” then by P(4), the probability of observing 4, we mean
P(A) = NẠ/N. (6.1)
Our defnition requires several comments. FEirst of all, we may speak of a
probability of something happening only if the occurrenece is a possible outcome
of some repeafabie observation. It is not clear that ít would make any sense to
ask: “What is the probability that there is a ghost in that house?”
You may object that no situation is ezacfly repeatable. That is right. Every
diferent observation must at least be at a diferent tỉme or place. All we can say
1s that the “repeated” observations should, for our intended purposes, øøøear
to be cquiuadlent. We should assume, at least, that each observation was made
from an equivalentÌy prepared situation, and especially with the same degree of
ignorance at the start. (If we sneak a look at an opponent°s hand in a card game,
our estimate of our chances of winning are different than if we do notÏ)
W©e should emphasize that NÑ and N4 in Eq. (6.1) are of intended to rep-
resent numbers based on actual observations. a4 is our best esfma#e of what
tuould occur in Ý ?magined observations. Probability depends, therefore, on our
knowledge and on our ability to make estimates. In efect, on our common sensel
Fortunately, there is a certain amount of agreement in the common sense oÝ many
things, so that diferent people will make the same estimate. Probabilities need
not, however, be “absolute” numbers. Since they depend on our ignorance, they
may become different if our knowledge changes.
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You may have noticed another rather “subJective” aspect of our defnition of
probability. We have referred to a as “our estimate of the most likely number
...” We do not mean that we expect to observe ezøctu Na, but that we expect
a number øcar Na, and that the number WA is more lkelu than any other
number in the vicinity. lf we toss a coin, say, 30 times, we should expect that the
number of heads would not be very likely to be exactly 15, but rather only some
number near to 1ð, say 12, 13, 14, 15, 16, or 17. However, if we must choose,
we would decide that 15 heads is more l2kely than any other number. We would
write P(heads) = 0.5.
Why did we choose 15 as more likely than any other number? We must have
argued with ourselves in the following manner: lf the most likely number of heads
1s Nụ in a total number of tosses , then the most likely number of tails M„~
1s (N — NH). (WG are assuming that every toss gives e/ther heads ør tails, and
no “other” resultl) But if the coin is “honest,” there is no preference for heads or
tails. Until we have some reason to think the coin (or 6oss) is dishonest, we musb
give equal likelihoods for heads and tails. 5o we must set W_- = Nh. It follows
that Nr = Nụ = N/2, or P(H) = P(T) = 05.
W© can generalize our reasoning to ømw situation in which there are w different
but “equivalent” (that is, equally likely) possible results of an observation. TÍ
an observation can yield mm. diferent results, and we have reason to believe that
any one of them is as likely as any other, then the probability of a particular
outcome 4 is P(4) = 1/m.
Tf there are seven diferent-colored balls in an opaque box and we pick one
out “at random” (that is, without looking), the probability of getting a ball
of a particular color 1s #- The probability that a “blind draw” from a shuffled
deck of 52 cards will show the ten of hearts is sB- The probability of throwing a
double-one with dice is z..
In Chapter 5 we described the size of a nucleus in terms of is apparent area, or
“cross section.” When we did so we were really talking about probabilities. When we
shoot a high-energy particle at a thin slab of material, there is some chance that it will
pass right through and some chance that it will hit a nucleus. (Since the nucleus is so
small that we cannot see it, we cannot aim right at a nucleus. We must “shoot blind”)
Tf there are m atoms in our slab and the nucleus of each atom has a cross-sectional
area ø, then the total area “shadowed” by the nuclei is nơ. In a large number of
random shots, we expect that the number of hits )œ of some nucleus will be in the
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ratio to /N as the shadowed area is to the total area of the slab:
NGƒN = nơ/A. (6.2)
We may say, therefore, that the probab7lztụ that any one projectile particle will sufer a
collision in passing through the slab is
Pe= T5: (6.3)
where ?+/A4 is the number of atoms per unit area in our slab.
6-2 Fluctuations
We would like now to use our ideas about probability to consider in some
greater detail the question: “How many heads do I really ezpec£ to get If Ï toss a
coin Ñ times?” Before answering the question, however, let us look at what does
happen in such an “experiment.” Figure 6-1 shows the results obtained in the first
three “runs” of such an experiment in which = 30. 'The sequences of “heads”
and “tails” are shown just as they were obtained. 'Phe first game gave 11 heads;
the second also 11; the third 16. In three trials we did not once get 15 heads.
Should we begin to suspect the coin? Or were we wrong in thinking that the
most likely number of “heads” in such a game is 15? NÑinety-seven more runs
were made to obtain a total of 100 experiments of 30 tosses each. The results of
the experiments are given in Table 6-1.
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