Probability Question

Question

Five cards are picked at random from a standard deck of 52 cards. For each of the following events, determine the number of sample points in the event. Leave your answer in symbolic form of combinations.

1. There are exactly 2 hearts in the 5 cards
2. There is at most 1 heart in the 5 cards
3. There is at least 1 heart in the 5 cards

Answer 1

Especially given that we are not required to solve, there are many alternate expressions. However, I will give the same answer as if I were computing.

Note throughout this problem that there are 52 cards in the deck, 13 hearts and 52-13=39 non-hearts.

a) 2 hearts in 5 cards means 2 of 13 hearts and 5-2 = 3 of 39 non-hearts

C(13,2)*C(39,3)

b) At most 1 heart means 1 or 0 hearts

0 hearts means 5 non-hearts, 1 heart 4 non-hearts

C(39,5) + C(39,4)C(13,1)

c) At least 1 heart in 5 cards means not just 0 hearts of 5,

As there are a total of C(52,5) hands possible, not just 0 hearts of 5 or not just 5 of 5 non-hearts is

C(52,5) - C(39,5)