Q. 4.31 - A First Course in Probability

Question

A jar contains $m+n$ chips, numbered $1,2,\dots ,n+m$. A set of size $n$ is drawn. If we let $X$ denote the number of chips drawn having numbers that exceed each of the numbers of those remaining, compute the probability mass function of $X$.

Step-by-Step Solution

Answer

$P(X=k)=\frac{\left(\begin{array}{c}n+m-k-1\\ n-k\end{array}\right)}{\left(\begin{array}{c}n+m\\ n\end{array}\right)}$

Step 1 Given information

A jar contains $m+n$ chips, numbered $1,2,\dots ,n+m$. A set of size $n$ is drawn. 

Step 2 Explanation

Observe that $X\in \{0,\dots ,n\}$. Take any $k\in \{0,\dots ,n\}$. Let's calculate $P(X=k)$. Observe that there are $\left(\begin{array}{c}n+m\\ n\end{array}\right)$ of all possible combinations of taken chips. If $X=k$, that means that we have taken $k$ largest number, have not taken $(k+1) s t$ largest number and all other remaining $n-k$ numbers out of remaining $n+m-k-1$ numbers have been taken freely. 

Step 3 Final answer

The probability mass function is

$P(X=k)=\frac{\left(\begin{array}{c}n+m-k-1\\ n-k\end{array}\right)}{\left(\begin{array}{c}n+m\\ n\end{array}\right)}$