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