Q.26 Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that: (i) all the five cards are spades. (ii) none is spade.
Best answer
Answer: (i) \(\dfrac{1}{1024}\) (ii) \(\dfrac{243}{1024}\)
Since the draws are with replacement, they are independent trials. Let \(X\) = number of spades; \(p = \dfrac{13}{52} = \dfrac14\), \(q = \dfrac34\), \(n = 5\). So \(X \sim B\left(5, \frac14\right)\) and \(P(X = x) = {}^5C_x\left(\frac14\right)^x\left(\frac34\right)^{5-x}\).
(i) \(P(X = 5) = \left(\dfrac14\right)^5 = \dfrac{1}{1024}\).
(ii) \(P(X = 0) = \left(\dfrac34\right)^5 = \dfrac{243}{1024}\).
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