A deck of 52 cards is mixed well, and 5 cards are dealt. a) It can be shown that
ID: 2959885 • Letter: A
Question
A deck of 52 cards is mixed well, and 5 cards are dealt.a) It can be shown that (disregarding the order in which the cards are dealt) there are 2,598,960 possible hands, of which only 1287 are hands consisting entirely of diamonds.
- what is the probability that a hand will consist entirely of diamonds?
- What is the probability that a hand will consist entirely of a single suit?
b) It can be shown that exactly 63,206 hands contain only clubs and diamonds, with both suits represented.
- What is the probability that a hand consists entirely of clubs and diamonds with both suits represented?
c) Using the result of Part b), what is the probability that a hand contains cards from exactly two suits?
Explanation / Answer
The probability of an event is the number of ways the event can happen divided by the total number of possible events, so the probability that a hand will consist entirely of diamonds is 1287/2598960 = 33/66640 = .000495
The probability that a hand will consist entirely of a single suit will be the sum of the probabilities that it will consist entirely of any particular suit, so it will be .000495 times 4 = .00198
The probability that a hand consists entirely of clubs and diamonds with both suits represented is 63206/2598960 = 143/5880 = .0243
For the probability that a hand will contain cards from exactly two suits, we first note that there will be no overlap among the sets of cards which contain exactly two suits and then count the number of combinations of two suits. There are 6 ways to pick 2 things out of 4, so the probability that a hand will contain cards from exactly two suits will be .0243 * 6 = .150