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Probability

Question
CBSEENMA12033856
Wired Faculty App

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(spade) = 13 over 52 space equals space 1 fourth comma space space straight q space equals space 1 minus straight p space equals space 1 minus 1 fourth space equals space 3 over 4
(i) P(5 spades) = straight C presuperscript 5 subscript 5 space open parentheses 1 fourth close parentheses to the power of 5 space equals space 1 cross times 1 over 1024 space equals space 1 over 1024
(ii) P(3 spades) = straight C presuperscript 5 subscript 3 space open parentheses 3 over 4 close parentheses squared space open parentheses 1 fourth close parentheses cubed space equals space straight C presuperscript 5 subscript 2 space cross times 9 over 16 cross times 1 over 64 space equals space fraction numerator 5 space cross times space 4 over denominator 1 space cross times 2 end fraction cross times 9 over 16 cross times 1 over 64 space equals 45 over 512
(iv) P(none  is a spade) = 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 space equals space 1 space cross times space 243 over 1024 space equals space 243 over 1024

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