Question

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?

Solution

Here n = 5
p = P(a spade in a single draw) = $13 over 52 equals space 1 fourth$
$therefore space space space space straight q space equals space 1 minus space straight p space equals space 1 minus space 1 fourth space equals space 3 over 4$
(i) P(all the five cards are spades)
$equals straight P left parenthesis 5 right parenthesis space equals space straight C presuperscript 5 subscript 5 space open parentheses 3 over 4 close parentheses to the power of 5 minus 5 end exponent space open parentheses 1 fourth close parentheses to the power of 5 space equals space 1 space cross times space 1 space cross times space 1 over 1024 space equals space 1 over 1024$
(ii) P (only 3 cards are spades)
$equals space straight P left parenthesis 3 right parenthesis space equals space straight C presuperscript 5 subscript 3 space open parentheses 3 over 4 close parentheses to the power of 5 minus 3 end exponent space open parentheses 1 fourth close parentheses cubed space equals space fraction numerator 5 space cross times space 4 space cross times 3 over denominator 1 space cross times 2 space cross times 3 end fraction cross times 9 over 16 cross times 1 over 64 space equals space fraction numerator 90 over denominator 16 space cross times space 64 end fraction space equals space 45 over 512$

(iii) P (none is a spade)
$equals straight P left parenthesis 0 right parenthesis space equals space straight C presuperscript 5 subscript 0 space open parentheses 3 over 4 close parentheses to the power of 5 minus 0 end exponent space space open parentheses 1 fourth close parentheses to the power of 0 space equals space 1 cross times 243 over 1024 cross times 1 space equals space 243 over 1024$

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Probability

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?

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For Daily Free Study Material Join wiredfaculty

Probability

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?

Some More Questions From Probability Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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?