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

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#### Solution

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

∴ n(S) = 52

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

Let A be the event that a red card is drawn

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

∴ n(A) = 26

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

Let B be the event that 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) = 26

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

There are 6 red face cards.

∴ n(A ∩ B) = 6

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

∴ Required probability = P(A ∪ B)

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

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

= `32/52`

= `8/13`

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