Question

You are dealt a hand of three​ cards, one at a time. Find the probability of each of the following.

​a) The first club you get is the third card dealt.

​b) Your cards are all spades.

​c) You get no black cards.

​d) You have at least one heart.

Answer 1

In general there are 26 red and 26 black cards in a pack. In 26 red cards, 13 heart cards and 13 diamond cards. In 26 black cards, 13 club cards and 13 spade cards. Total number of cards in a pack is 52. In this case, it can be observed that first two cards are not club cards. That is, there are 39 =52 - 13) cards are favourable to draw first two cards. The required probability is obtained as follows: Required Probability = P(First card) P(Second card) P(Third card) 39 38 13 52 51 50 =0.1453 In this case, the spade cards are 13 and total number of cards is 52. The required probability is obtained as follows: 13 12 11 P(All spades) == = = 52 51 50 = 0.0129

In this case, the 26 black cards are not favourable to draw. The required probability is obtained as follows: P(No black cards) = P(All red cards) _26 25 24 52 51 50 =0.1176

In this case, the more than one card is heart. The required probability is obtained as follows: P(Atleast one heart)=1-P(No heart) =1-39 38 37 =1-52 51 50 =1-0.4135 = 0.5865