Sum

Four cards are drawn from a pack of 52 cards. Find the probability that at least one heart.

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#### Solution

4 cards can be drawn out of 52 cards in ^{52}C_{4} ways.

∴ n(S) = ^{52}C_{4 }

Let C be the event that out of the four cards drawn at least one is a heart.

∴ C′ is the event that all 4 cards drawn are non-heart cards.

In a pack of 52 cards, there are 39 non-heart cards.

∴ 4 non-heart cards can be drawn in ^{39}C_{4} ways.

∴ n(C′) = ^{39}C_{4}

∴ P(C′) = `("n"("C'"))/("n"("S")` = `(""^39"C"_4)/ (""^52"C"_4)`

∴ P(C) = 1 – P(C') = `1 - (""^39"C"_4)/(""^52"C"_4)`

Concept: Elementary Properties of Probability

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