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Question
Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards; find the probability that none is a spade.
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Solution
Let X denote number of spade cards.
p = probability of drawing a spade card from 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(none of cards is spade):
P(X = 0) = P(0) = `"^5C_0(1/4)^0(3/4)^(5 - 0)`
`= 1xx1 xx (3/4)^5`
= `243/1024`
Hence, the probability of none of the spade card is `243/1024`.
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