5 cards are drawn successively from a well-shuffled pack of 52 cards with replacement. Determine the probability that (i) all the five cards should be spades? (ii) only 3 cards should be spades? (iii) none of the cards is a spade?
Solution:
Let us assume that X be the number of spade cards
Using the Bernoulli trial, X has a binomial distribution
P(X = x) = nCx qn-x px
Thus, the number of cards drawn, n = 5
Probability of getting spade card, p = 13/52 = 1/4
Thus the value of the q can be found using
q = 1 – p = 1 – (1/4)= 3/4
Now substitute the p and q values in the formula,
Hence, P(X = x) = 5Cx (3/4)5-x(1/4)x
(1) Probability of Getting all the spade cards:
P(all the five cards should be spade) = 5𝐶5 (1/4)5(3/4)0
= (1/4)5
= 1/1024
(2) Probability of Getting only three spade cards:
P(only three cards should be spade) = 5𝐶3 (1/4)3(3/4)2
= (5!/3! 2!) × (9/1024)
= 45/ 512
(3) Probability of Getting no spades:
P(none of the cards is a spade) = 5𝐶0(1/4)0(3/4)5
= (3/4)5
= 243/ 1024