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?

Answer

N/a

Explanation

Let `X` represents the numbers of spades in 5 cards drown. There are 13 cards of spades in a pack of 52 cards.
`p=P` (getting a card of spade)`=13/52=1/4`
and `q=1-1/4=3/4`
`:.X` is a binomial distribution
where `n=5,p=1/4` and `q=3/4`
Therefore, `P(X=r)=.^(n)C_(r)p^(r)q^(n-r)`,
where `r=0,1,2,...........n`
`P(X=r)=.^(5)C_(r).(1/4)^(r)(3/4)^(5-r)`
(Using binomial distribution)
(i) `P` (getting all 5 cards of spade) `=P(X=5)`
`=.^(5)C_(5)p^(5)q^(0)=1.p^(5)=(1/4)^(4)=1 . 1/1024=1/1024`
(ii) `P` (getting exactly 3 cards of spades) `=P(X=3)`
`.^(5)C_(3)p^(3)q^(2)=(5.4.3)/(3!)(1/4)^(3).(3/4)^(2)`
`=60/(1xx2xx3)xx(3^(2))/(4^(5))=90/1024=45/512`
(iii) `P` (no card of spade is obrained)
`P(X=0)=.^(5)C_(0)p^(0)q^(5)=(3/4)^(5)=243/1024`