Direct reasoning and indirect reasoning in math-2 alberta curriculum

Question

A standard deck of 52 cards consists of 4 suits (hearts, clubs, diamonds and spades). If a student deals 3 cards from standard deck, determine the probability to the nearest hundredth, that at least one card is a heart.

Use method stated below:

a) direct reasoning:

b) indirect reasoning:

Answer 1

a) If you have at least one heart in a selection of 3 cards, then you have one, two, or three hearts.

P(1 heart in a selection of 3 cards) = C(13,1)C(39,2)/C(52,3)

P(2 hearts in a selection of 3 cards) = C(13,2)C(39,1)/C(52,3)

P(3 hearts in a selection of 3 cards) = C(13,3)/C(52,3)

P(at least one heart) = (C(13,1)C(39,2) +C(13,2)C(39,1)+C(13,3))/C(52,3) = 12961/22100 =997/1700 = .5865

b) If you don't have all non-hearts, then you have at least one heart.

P(not having all non-hearts) = 1 - P(having all non-hearts) = 1 - C(39,3)/C(52,3) = 1 - 703/1700 = 997/1700 = .5865.