Question

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

Answer 1

4 cards can be drawn out of 52 cards in ⁵²C₄ ways.
∴ n(S) = ⁵²C₄ 
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 ³⁹C₄ ways.
∴ n(C′) =  ³⁹C₄

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