Q.5 (B) (iii) Five cards are drawn successively with replacement from a well shuffled deck of 52 cards. Find the probability that :
(a) all the five cards are spades
(b) only 3 cards are spades.
Best answer
Answer: (a) \(\frac{1}{1024}\) (b) \(\frac{45}{512}\)
Draws are with replacement, so they are independent trials. Let X = number of spades; \(X\sim B\left(5,\frac{1}{4}\right)\), with \(p=\frac{13}{52}=\frac{1}{4},\ q=\frac{3}{4}\).
\(P(X=x)={}^5C_x\left(\frac{1}{4}\right)^x\left(\frac{3}{4}\right)^{5-x}\)
(a) \(P(X=5)=\left(\frac{1}{4}\right)^5=\frac{1}{1024}\)
(b) \(P(X=3)={}^5C_3\left(\frac{1}{4}\right)^3\left(\frac{3}{4}\right)^2=10\times\frac{1}{64}\times\frac{9}{16}=\frac{90}{1024}=\frac{45}{512}\)
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