Chapter 2.5, Problem 11E

A Bingo card has 25 squares with numbers on 24 of them, the center being a free square. The integers that are placed on the Bingo card are selected randomly and without replacement from 1 to 75, inclusive. When a game called “cover-up” is played, balls numbered from 1 to 75, inclusive, are selected randomly and without replacement until a player covers each of the numbers on a card. Let X equal the number of balls that must be drawn to cover all the numbers on a single card.

(a) Argue that the pmf of X, for $x=24,25,\mathrm{...},75$, is

$\begin{array}{l}f\left(x\right)=\frac{\left(\begin{array}{c}24\\ 23\end{array}\right)\left(\begin{array}{c}51\\ x-24\end{array}\right)}{\left(\begin{array}{c}75\\ x-1\end{array}\right)}•\frac{1}{75-\left(x-1\right)}\\ =\frac{\left(\begin{array}{c}51\\ x-24\end{array}\right)}{\left(\begin{array}{c}75\\ x\end{array}\right)}•\frac{24}{x}=\frac{\left(\begin{array}{c}x\\ 24\end{array}\right)}{\left(\begin{array}{c}75\\ 24\end{array}\right)}•\frac{24}{x}.\end{array}$

(b) What value of X is most likely to occur? In other words, what is the mode of this distribution?

(c) To show that the mean of X is $\frac{\left(24\right)\left(76\right)}{25}=72.96$ use the combinatorial identity $\left(\begin{array}{c}k+n+1\\ k+1\end{array}\right)={\displaystyle \underset{x=k}{\overset{n+k}{\sum }}\left(\begin{array}{c}x\\ k\end{array}\right)}$.

(d) Show that $E\left[X\left(X+1\right)\right]=\frac{24•77•76}{26}=5401.8462$

(e) Calculate $\mathrm{var}\left(x\right)=E\left[X\left(X+1\right)\right]-E\left[X\right]-E{\left[X\right]}^2$ $=\frac{46,516}{8125}=5.7246$ and $\sigma =2.39$.