Question

A card from a pasck of 52 card is lost.From the remaining cards of the pack,two cards are drawn and are found to be both diamonds.Find the probability of the lost card being a diamond.

Answer 1

Let E1=the missing card is a diamond,E2=the missing card is a spade,E3=the missing card is a club,E4=the missing card is a heart and A=drawing of two heart cards from the remaining cards.
Now P(E1)=$\frac{13}{52}$=$\frac{1}{4}$,P(E2)=$\frac{13}{52}$=$\frac{1}{4}$,P(E3)=$\frac{13}{52}$=$\frac{1}{4}$,P(E4)=$\frac{13}{52}$=$\frac{1}{4}$
P(A|E1)=P(drawing 2 heart cards given that one diamond card is missing)=$\frac{C(12,2)}{C(51,2)}$
By Bayes' theorem,
$P(E_1|A)=\frac{P(E_1)P(A|E_1)}{P(E_1)P(A|E_1)+P(E_2)P(A|E_2)+P(E_3)P(A|E_3)+P(E_4)P(A|E_4)}$
$=\frac{1}{4}\times \frac{C(12,2)}{C(51,2)}$
=$\frac{66}{66}$+78+78+7$$8=$\frac{11}{50}$