Question

Given a standard deck of cards, and X = the number of King or Queens picked. Suppose four cards are picked, give the distribution of the probabilities of X = 0,1,2,3,4 the variance and the standard deviation with and without replacement of the chosen card.

Answer 1

a)here with replacement:

this follows binomial distribution with parameter n=4 and p=8/52 =2/13

P(X=x)=⁴C_x(2/13)^x(11/13)^4-x

from above:

x	P(x)
0	0.5126
1	0.3728
2	0.1017
3	0.0123
4	0.0006

mean =np=4*(2/13)=8/13=0.615

variance=(np(1-p)) =0.5207

standard deviation =sqrt(np(1-p)) =0.7216

2)without replacement:

this follows Hyperegeometric distribution with parameter n=4 ; k=8 ; N=52 (As there are 8 cards of King or Queen among 52 cards)

P(X=x)=⁸C_x⁴⁴C_4-x /⁵²C₄

x	P(x)
0	0.5014
1	0.3914
2	0.0978
3	0.0091
4	0.0003

mean =nk/N =0.6154

variance =nk/N*(1-k/N)*(N-n)/(N-1)=0.49008

standard deviation =sqrt(0.49008)=0.700057