Ex 13.5, 4

Transcript

Ex 13.5, 4 Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that (i) all the five cards are spades? (ii) only 3 cards are spades? (iii) none is a spade? Let X : be the number of spade cards Drawing is a Bernoulli trial So, X has binomial distribution P(X = x) = nCx 𝒒﷮𝒏−𝒙﷯ 𝒑﷮𝒙﷯ n = number of cards drawn = 5 p = Probability of getting spade card = 13﷮52﷯= 1﷮4﷯ q = 1 – p = 1 – 1﷮4﷯= 3﷮4﷯ Hence, P(X = x) = 5Cx 𝟑﷮𝟒﷯﷯﷮𝟓−𝒙﷯ 𝟏﷮𝟒﷯﷯﷮𝒙﷯ • P(all cards are spade) = 5𝐶5 1﷮4﷯﷯﷮5﷯ 3﷮4﷯﷯﷮0﷯ = 1﷮4﷯﷯﷮5﷯ = 𝟏﷮𝟏𝟎𝟐𝟒﷯ • P(only three cards are spade) = 5𝐶3 1﷮4﷯﷯﷮3﷯ 3﷮4﷯﷯﷮2﷯ = 5!﷮3! 2!﷯ × 9﷮1024﷯ = 𝟒𝟓﷮𝟓𝟏𝟐﷯ (iii) P(none of them are spade) = 5𝐶0 1﷮4﷯﷯﷮0﷯ 3﷮4﷯﷯﷮5﷯ = 3﷮4﷯﷯﷮5﷯ = 𝟐𝟒𝟑﷮𝟏𝟎𝟐𝟒﷯