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Question
Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards; find the probability that all the five cards are spades.
Solution
Let X denote number of spade cards.
p = probability of drawing a spade card from a pack of 52 cards.
Since there are 13 spade cards in the pack of 52 cards,
∴ p = `13/52 = 1/4` and q = 1 − p = `1 - 1/4 = 3/4`
Given, n = 5
∴ `X ~ B(5, 1/4)`
The p.m.f. of X is given by
P(X = x) = `"^nC_x p^x q^(n - x)`
i.e. p(x) = `"^5C_x (1/4)^x (3/5)^(5 - x)`, x = 0, 1, 2, ..., 5
P(all five cards are spades):
= P(X = 5) = p(5) = `"^5C_5(1/4)^5(3/4)^(5 - 5)`
`= 1(1/4)^5(3/4)^0`
`= 1 xx 1/1024 xx 1`
= `1/1024`
= `1/4^5`
Hence, the probability of all the five cards are spades is `1/4^5`.
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