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# Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that  none is a spade ? - Mathematics

Sum

Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that  none is a spade ?

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#### Solution

Let X denote the number of spade cards when 5 cards are drawn with replacement.  Because it is with replacement,

X follows a binomial distribution with n = 5; $p = \frac{13}{52} = \frac{1}{4}; q = 1 - p = \frac{3}{4}$

$P(X = r) = ^{5}{}{C}_r \left( \frac{1}{4} \right)^r \left( \frac{3}{4} \right)^{5 - r}$
$P(\text{ none is a spade } ) \hspace{0.167em} = P(X = 0)$
$= ^{5}{}{C}_0 \left( \frac{1}{4} \right)^0 \left( \frac{3}{4} \right)^5$
$= \frac{243}{1024}$
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 33 Binomial Distribution
Exercise 33.1 | Q 12.3 | Page 13
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