Chapter 2.1, Problem 15E

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:

2 1 2 1 0 0 1 0 1 1 0 2 0 2 3 0 1 1 0 3

1 2 0 2 0 2 0 1 0 1 1 2 1 0 1 1 2 1 1 0

(a) Argue that the pmf of X is $f\left(x\right)=\frac{\left({}_x^{12}\right)\left({}_{5-x}^{40}\right)}{\left({}_5^{52}\right)},x=0,1,2,3,4,5,$ and thus, that $f\left(0\right)=2109/8330,f\left(1\right)=703/1666,f\left(2\right)=209/833,f\left(3\right)=55/833,f\left(4\right)=165/21,658,$ and $f\left(5\right)=33/108,290.$

(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.