Five cards are drawn successively with replacement from a well-shuffle

1.	Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that (i) all the five cards are spades? (ii) only 3 cards are spades? (iii) none is a spade?
Answer» Let X represent the number of spade cards among the five cards drawn. Since, the drawing card is with replacement, the trials are Bernoulli trials. In a well-shuffled deck of 52 cards, there are 13 spade cards. p = P (success) = P (a spade card is drawn) = 13/52 = 1/4 and q = 1 - p = 1 - 1/4 = 3/4 X has a binomial distribution with n = 5, p = 1/4 and q = 3/4 Therefore, by Binomial distribution P(X = r) = ⁿC_r p^rq^{n – r}, where r = 0, 1, 2,...,n P(X = r) = ⁵C_r(1/4)^r(3/4)^{5 - r} (i) P (all the five cards are spades) = P( X = 5) = ⁵C₅P⁵q⁰ = P⁵= (1/4)⁵ = 1/1024 (ii) P (only three cards are spades) = P( X = 3) = ⁵C₃P³q² = 10(1/4)³(3/4)² = 90/1024 = 45/512 (iii) P (none is a spade) = P (X = 0) = ⁵C₀P⁰q⁵= (3/4)⁵ = 243/1024

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that (i) all the five cards are spades? (ii) only 3 cards are spades? (iii) none is a spade?

Answer»

Let X represent the number of spade cards among the five cards drawn. Since, the drawing card is with replacement, the trials are Bernoulli trials.

In a well-shuffled deck of 52 cards, there are 13 spade cards.

p = P (success) = P (a spade card is drawn) = 13/52 = 1/4

and q = 1 - p = 1 - 1/4 = 3/4

X has a binomial distribution with n = 5, p = 1/4 and q = 3/4

Therefore, by Binomial distribution

P(X = r) = ⁿC_r p^rq^{n – r}, where r = 0, 1, 2,...,n

P(X = r) = ⁵C_r(1/4)^r(3/4)^{5 - r}

(i) P (all the five cards are spades) = P( X = 5) = ⁵C₅P⁵q⁰ = P⁵= (1/4)⁵ = 1/1024

(ii) P (only three cards are spades) = P( X = 3) = ⁵C₃P³q² = 10(1/4)³(3/4)² = 90/1024 = 45/512

(iii) P (none is a spade) = P (X = 0) = ⁵C₀P⁰q⁵= (3/4)⁵ = 243/1024