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5 cards are drawn successively with replacement from 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.

2 Answers

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Probability of x success in n Bernoulli trials is  $P(X=x)=C(n,x)p^{x}q^{n-x}$

p=probability of spade drawn = $\frac{13}{52}=\frac{1}{4}$

q=probability of spade not drawn = $1- \frac{1}{4} = \frac{3}{4}$

i) P(all five cards are spade) = $P(X=5)=C(5,5)(\frac{1}{4})^{5}(\frac{3}{4})^{0} = \frac{1}{1024}$

ii) P(only 3 cards are spade) =$P(X=3)=C(5,3)(\frac{1}{4})^{3}(\frac{3}{4})^{2} = \frac{90}{1024}$

iii) P(none is spade) = $P(X=0)=C(5,0)(\frac{1}{4})^{0}(\frac{3}{4})^{5} = \frac{243}{1024}$
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i) all the five cards are spades with replacement=$\frac{13^{5}}{52^{5}}$

ii) only 3 cards are spades =$\frac{13^{3}\times 39^{2}}{52^{5}}$

iii) none is a spade=$\frac{39^{5}}{52^{5}}$

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