Three cards are drawn at random (without replacement) from a a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence find the mean of the distribution.

Answer

Mean, `E(X)=1.5`

Explanation

Let `X` denote the number of red cards. Three cards are drawn from a pack of 52 cards.


`therefore X "can take values" 0,1,2,3`


Now the probability of drawing red cards are:


` P(X=0)=P("no red card drawn")`


`=(26C_0 times 26C_3)/(52C_3)=(frac{26!}{0!26!} times frac{26!}{3!23!})/(frac{52!}{3!49!})`


`=(frac{26!}{1 times 26!} times (26 times 25 times 24 times 23!)/(3 times 2 times 23!))/((52 times 51 times 50 times 49!)/(3 times 2 times 49!))=frac{26 times 25 times 24}{52 times 51 times 50}=2/17`


`rArr P(X=0)=2/17`


` P(X=1)=P("one red card drawn")`


...