A card is drawn from a pack of 52 cards. What is the probability that, card is either black or a face card?

#### Solution

One card can be drawn from the pack of 52 cards in ^{52}C_{1} = 52 ways

∴ n(S) = 52

The pack of 52 cards consists of 26 red and 26 black cards.

Let event A: A black card is drawn

∴ Black card can be drawn in ^{26}C_{1} = 26 ways.

∴ n(A) = 26

∴ P(A) = `("n"("A"))/("n"("S")) = 26/52`

Let event B: A face card is drawn

There are 12 face cards in the pack of 52 cards.

∴ 1 face card can be drawn in ^{12}C_{1} = 12 ways.

∴ n(B) = 12

∴ P(B) = `("n"("B"))/("n"("S")) = 12/52`

There are 6 black face cards.

∴ n(A ∩ B) = 6

∴ P(A ∩ B) = `("n"("A" ∩ "B"))/("n"("S")) = 6/52`

∴ Required probability = P (A ∪ B)

= P(A) + P(B) – P(B) – P(A ∩ B)

= `26/52 + 12/52 - 6/52`

= `32/52`

= `8/13`