Question

Five cards are drawn one by one, with replacement, from a well-shuffled deck of 52 cards. Find the probability that
(i) all the five cards diamonds
(ii) only 3 cards are diamonds
(iii) none is a diamond

Answer 1

Let the random variable X = no. of diamonds
So, X can take values 0,1,2,3,4 and 5.
Also, p = P(success) = P(a diamond) = \[\frac{13}{52} = \frac{1}{4}\] And q = P(failure) = \[1 - p\] = \[1 - \frac{1}{4} = \frac{3}{4}\]

Now,

\[\left( i \right) P\left( X = 5 \right) = ^{5}{}{C}_5 p^5 = \left( \frac{1}{4} \right)^5 = \frac{1}{1024}\]

\[\left( ii \right) P\left( X = 3 \right) = ^{5}{}{C}_3 p^3 q^2 = 10 \left( \frac{1}{4} \right)^3 \left( \frac{3}{4} \right)^2 = \frac{90}{1024} = \frac{45}{512}\]

\[\left( iii \right) P\left( X = 0 \right) = ^{5}{}{C}_0 q^5 = \left( \frac{3}{4} \right)^5 = \frac{243}{1024}\]