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A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 2: Q. 2.51 (page 51)

Suppose that $n$ balls are randomly distributed into $N$ compartments. Find the probability that $m$ balls will fall into the first compartment. Assume that all $N^{n}$ arrangements are equally likely.

Short Answer

Expert verified

$(\begin{matrix} n \\ m \end{matrix}) {(N - 1)}^{n - m}$

Step by step solution

01

Given Information.

Suppose that $n$ balls are randomly distributed into $N$ compartments.

02

Explanation.

$n$ balls.

$N$ compartments.

Each of the $N^{n}$ outcomes is equally likely

localid="1649038659363" $P r o b a b i l i t y o f A = e x a c t l y m b a l l s f a l l i n t o t h e f i r s t c o m p a r t m e n t$

The outcome space $S$ is a set of $n$ valued vectors where each element is from ${1, 2, 鈥�, N}$ and describes in which compartment did that ball went.

As these events are equally likely -

$A 鈯� S 鈫� P (A) = \frac{| A |}{| S |}$

There are $(\begin{array}{l} n \\ m \end{array})$ choices for the balls that fall into the first compartment. And for each choice of these balls, the rest of the balls $(n - m$ of them) have to be in some of the $N - 1$ remaining compartments.

$So | A | = (\begin{matrix} n \\ m \end{matrix}) (N - 1)^{n - m}$

$P (A) = \frac{| A |}{| S |} = \frac{(\begin{matrix} n \\ m \end{matrix}) (N - 1)^{n - m}}{N^{n}}$

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Most popular questions from this chapter

A group of individuals containing $b$ boys and $g$ girls is lined up in random order; that is, each of the $(b + g)!$ permutations is assumed to be equally likely. What is the probability that the person in the ith position, role="math" localid="1648906629368" $1 鈮� i 鈮� b + g,$ is a girl?

The following data were given in a study of a group of $1000$ subscribers to a certain magazine: In reference to the job, marital status, and education, there were $312$ professionals, $470$ married persons, $525$ college graduates, $42$ professional college graduates, $147$ married college graduates, $86$ married professionals, and $25$ married professional college graduates. Show that the numbers reported in the

the study must be incorrect.

Hint: Let $M, W,$ and $G$ denote, respectively, the set of professionals, married persons, and college graduates. Assume that one of the $1000$ persons is chosen at random, and use Proposition $4.4$ to show that if the given numbers are correct, then $P (M 鈭� W 鈭� G) > 1$ .

If it is assumed that all $(\begin{matrix} 52 \\ 5 \end{matrix})$ poker hands are equally likely, what is the probability of being dealt

$(a)$ a flush? (A hand is said to be a flush if all $5$ cards are of the same suit.)

$(b)$ one pair? (This occurs when the cards have denominations $a, a, b, c, d,$ where $a, b, c,$ and $d$ are all distinct.)

$(c)$ two pairs? (This occurs when the cards have denominations $a, a, b, b, c,$ where $a, b,$ and $c$ are all distinct.)

$(d)$ three of a kind? (This occurs when the cards have denominations $a, a, a, b, c,$ where $a, b,$ and $c$ are all distinct.)

$(e)$ four of a kind? (This occurs when the cards have denominations $a, a, a, a, b$ )

$(a)$ If $N$ people, including $A$ and $B$ , are randomly arranged in a line, what is the probability that $A$ and $B$ are next to each other?

$(b)$ What would the probability be if the people were randomly arranged in a circle?

Let $T_{k} (n)$ denote the number of partitions of the set $1, . . ., n$ into $k$ nonempty subsets, where $1 鈮� k 鈮� n$ . (See Theoretical Exercise $8$ for the definition of a partition.) Argue that

$T_{k} (n) = k T_{k} (n 鈭� 1) + T_{k} 鈭� 1 (n 鈭� 1)$

Hint: In how many partitions is $1$ a subset, and in how many $1$ elements of a subset that contains other elements?

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