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?

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Let us assume that X be the number of spade cards

Using the Bernoulli trial, X has a binomial distribution

P(X = x) =nCxqn-xpx

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

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