Q. 4.32 - A First Course in Probability

Question

A jar contains $n$ chips. Suppose that a boy successively draws a chip from the jar, each time replacing the one drawn before drawing another. The process continues until the boy draws a chip that he has previously drawn. Let $X$ denote the number of draws, and compute its probability mass function.

Step-by-Step Solution

Answer

$\frac{k}{n^{k+1}}·\prod _{i=0}^{k-1}(n-i)$

Step 1 Given information

We have that $X\in \mathrm{\mathbb{N}},X\ge 2$. Event $X>k$ means that in first $k$ draws we have taken all different chips. Hence

$P(X>k)=\frac{n(n-1)\dots (n-k+1)}{n^k}$

Similarly we have that 

$P(X>k+1)=\frac{n(n-1)\dots (n-k)}{n^{k+1}}$

Step 2 Explanation

Hence. we have that 

$P(X=k)=P(X>k)-P(X>k+1)$

$=\frac{n(n-1)\dots (n-k+1)}{n^k}-\frac{n(n-1)\dots (n-k)}{n^{k+1}}$

$=\frac{n(n-1)\dots (n-k+1)}{n^k}\left(1-\frac{n-k}{n}\right)$

$=\frac{k}{n}·\frac{n(n-1)\dots (n-k+1)}{n^k}$

Step 3 Final answer

The probability mass function is

$P(X=k)=\frac{k}{n^{k+1}}·\prod _{i=0}^{k-1}(n-i)$