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

Given Information.

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

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 ${1, 2, \dots, N}$ and describes in which compartment did that ball went.

As these events are equally likely -

$A \subseteq S \to P (A) = \frac{| A |}{| S |}$

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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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

Step by step solution

Given Information.

Explanation.

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Suppose thatnballs are randomly distributed into Ncompartments. Find the probability that mballs will fall into the first compartment. Assume that all Nnarrangements are equally likely.

Short Answer

Step by step solution

Given Information.

Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

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.