Two cards are drawn from a pack of 52 cards. What is the probability that, both the cards are either black or queens?

#### Solution

Two cards can be drawn from 52 cards in ^{52}C_{2 }ways.

∴ n(S) = `""^52"C"_2`

Also, the pack of 52 cards consists of 26 red and 26 black cards.

Let A be the event that both cards are black.

∴ 2 black cards can be drawn in ^{26}C_{2} ways.

∴ n(A) = ^{26}C_{2}

∴ P(A) = `("n"("A"))/("n"("S"))=(""^26"C"_2)/(""^52"C"_2)=(26xx25)/(52xx51)=25/102`

Let B be the event that both cards are queens. There are 4 queens in a pack of 52 cards

∴ 2 queen cards can be drawn in ^{4}C_{2} ways.

∴ n(B) = `""^4"C"_2`

∴ P(B) = `("n"("B"))/("n"("S")) =(""^4"C"_2)/(""^52"C"_2)= (4xx3)/(52xx51) = 1/221`

There are two black queen cards.

∴ n(A ∩ B) = `""^2"C"_2` = 1

∴ P(A ∩ B) = `("n"("A" ∩ "B"))/("n"("S")) = 1/(""^52"C"_2)=(1xx2xx1)/(52xx51)`

= `1/1326`

∴ Required probability = P(A ∪ B)

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

= `25/102 + 1/221 - 1/1326`

= `325/1326 + 6/1236 - 1/1326`

= `330/1326`

= `55/221`