Combinations of Married CouplesDate: 11/08/96 at 02:33:40 From: Jingde Chu Subject: Combinations of couples Given that there are 6 married couples: a. What is the probability that the 12 people can be grouped into 6 pairs where each pair is a married couple? b. If we select only 4 people out of the 12, what is the probability that: i. none of them are married couples? ii. there is only one pair of married couples? iii. there are two pairs of married couples? Date: 11/08/96 at 08:23:29 From: Doctor Donald Subject: Re: Combinations of couples >a. What is the probability that the 12 people can be grouped into 6 pairs where each pair is a married couple? Suppose the men are denoted by a, b, c, d, e, f and the women are denoted by A, B, C, D, E, F where a is married to A and so on. Starting with man a, there is 1/6 chance he is matched with his wife A. Move to man b, the chance that he is matched with B from the remaining 5 women is 1/5. Move to man c, the chance that he is matched with C is 1/4, and so on. The total probability that each pair is a married couple is: 1/6 x 1/5 x 1/4 x 1/3 x 1/2 x 1/1 = 1/720. >b. If we select only 4 people out of the 12, what is the probability that: i. none of them are married couples: Consider the couples a b c d e f A B C D E F First we select 4 columns, then one item from each column. The number of ways to select 4 from 6 columns is 6_C_4 = 15. The number of ways to select one member from the pair is 2_C_1 = 2. The number of ways to select 4 people, none of whom are a pair, is 15 x 2 = 30. The number of groups of 4 people chosen without restriction from 12 people is (1/3)12_C_4 = 165. The probability that there are no married couples is 30/165 = 2/11. ii. there is only one pair of a married couples: Now we must select one column from the six (in 6 ways) and then two other columns from the other 5. This can be done in 5_C_2 = 10 ways. Of these, there are two ways to select one member of the pair. The total number of ways to select just one pair is 6 x 10 x 2 = 120. The probability that there is only one pair of married couples is 120/165 = 8/11. iii. there are two pairs of married couples: We could get this from finding 1 - 2/11 - 8/11 = 1/11, but we can also calculate it directly. We must select 2 columns from the six. This can be done in 15 ways, and so the probability that there are two pairs of married couples is 15/165 = 1/11. -Doctor Donald, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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