Chapter 2.5, Problem 5E

Five cards are selected at random without replacement from a standard, thoroughly shuffled 52-card deck of playing cards. Let X equal the number of face cards (kings, queens. jacks) in the hand. Forty observations of X yielded the following data:

(a) Argue that the pmf of X is

$f\left(x\right)=\frac{\left(\begin{array}{c}12\\ x\end{array}\right)\left(\begin{array}{c}40\\ 5-x\end{array}\right)}{\left(\begin{array}{c}52\\ 5\end{array}\right)},x=0,1,2,3,4,5,$ and thus, that $f\left(0\right)=\frac{2109}{8330},f\left(1\right)=\frac{703}{1666},f\left(2\right)=\frac{209}{833},f\left(3\right)=\frac{55}{833},f\left(4\right)=\frac{165}{21,658},$ and $f\left(5\right)=\frac{33}{108,290}.$

(b) Draw a probability histogram for this distribution.

(c) Determine the relative frequencies of 0. 1, 2. 3, and superimpose the relative frequency histogram on your probability histogram.