Five cards are selected at random without replacement from a standard, thoroughly shuffled 52card 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
and thus, that f (0) = 2109/8330, f (1) = 703/1666, f (2) = 209/833, f (3) = 55/833, f (4) = 165/21,658, and f (5) = 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.
Answer :
Step 1of 3 :
Five cards are selected at random without replacement from a standard,thoroughly shuffled 52card deck of playing cards.
Let X equal the number of face cards (kings,queens,jacks ) in hand.
a).
There are 12 face cards among 52 cards.We can select the number of number of face cards in any one ways and 52 cards not face cards in any one ways, by the multiplication principle product is equals the number of ways the joint operation can he performed .
If we assume that each of ways of selecting 5 objects from 52 objects has the same probability.
Here X = number of face cards.
12 = total number of face cards.
40 = total number of non face cards.
Then,
f(x) = , x = 0,1,2,3,4,5
So put x = 0,1,2,3,4, and 5
Then,
f(0) =
f(0) = 0.2532
f(1) =
f(1) = 0.42197
f(2) =
f(2) = 0.251
f(3) =
f(3) = 0.066
f(4) =
f(4) = 0.00762 and
f(5) =
f(5) = 0.00030
Step 2 of 3 :
b).
We have drawn a probability for this distribution.
f(x) 
h(x) 

0 
0.2532 
0.325 
1 
0.4219 
0.4 
2 
0.251 
0.225 
3 
0.066 
0.05 
4 
0.0076 
0 
5 
0.0003 
0 
Then the graph is given below.