Question

Three cards are drawn at random successively, with replacement, from a well shuffled pack of 52 cards. Getting a card of diamond is termed as success. Obtain the probability distribution of the number of success.

Answer 1

Probability Distribution : 

1) Let X denote the no. of diamond cards in three trials With Replacement and X = 0,1,2,3 .
The, P(X=0) =  Probabilty of not getting diamond in all three trials .
Similarly ,we can define P(X=1) &P(x=2) &P(X=3).

Probabilty of getting a diamond card ,P(Success,S) = $\frac{13}{52}$
=> P(S) = $\frac{1}{4}$
Probability of failing or not getting diamond card,
P(F) = $1- \frac{1}{4} = \frac{3}{4}$

We also need to take care of order.
 2) P(X=0) = Three fails
                 = P(F)*P(F)*P(F)  *$\frac{3!}{3!}$
                 = $\frac{3}{4}*\frac{3}{4} *\frac{3}{4} =\frac{27}{64}$

3) P(X=1)= One Success Two Fails 
                =  P(S)*P(F)*P(F) *$\frac{3!}{2!}$
               =$\frac{1}{4}*\frac{3}{4} *\frac{3}{4} *3=\frac{27}{64}$

4) P(X=2) = Two Success One Fail 
                 = $\frac{1}{4} *\frac{1}{4} *\frac{3}{4} *\frac{3!}{2!}=\frac{9}{64}$  

5) P(X=3) = Three Success  
                 = $\frac{1}{4} *\frac{1}{4} *\frac{1}{4} *\frac{3!}{3!} = \frac{1}{64}$