I have a probability question from the book Fundamentals of Probability by Saeed Ghahramani. The question is "Suppose that 12 married couples take part in a contest. If 12 persons each win a prize, what is the probability that from every couple one of them is a winner? Assume that all of the (24 12) possible sets of winners are equally probable." (24 12) means 24 total choices, choose 12. Order does not matter. I have the answer, it was in the back, but I would like to understand how to do it. Thank you very much!
PS answer is rounded to .00151 in case you were wondering.
PS answer is rounded to .00151 in case you were wondering.
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Imagine the people drawing tickets, one at a time, husband then wife:
Mr. A can draw anything
Mrs. A must draw the opposite, i.e. out of 23 remaining tickets, 12 are the opposite of what Mr. A drew, so probability 12/23
Mr. B can draw anything
Mrs. B must draw the opposite, i.e. out of 21 remaining tickets, 11 are the opposite of what Mr. B drew, so probability 11/21
.... and so on
So probability is:
12/23 x 11/21 x 10/19 x 9/17 x 8/15 x 7/13 x 6/11 x 5/9 x 4/7 x 3/5 x 2/3 x 1/1
= 0.00151
Mr. A can draw anything
Mrs. A must draw the opposite, i.e. out of 23 remaining tickets, 12 are the opposite of what Mr. A drew, so probability 12/23
Mr. B can draw anything
Mrs. B must draw the opposite, i.e. out of 21 remaining tickets, 11 are the opposite of what Mr. B drew, so probability 11/21
.... and so on
So probability is:
12/23 x 11/21 x 10/19 x 9/17 x 8/15 x 7/13 x 6/11 x 5/9 x 4/7 x 3/5 x 2/3 x 1/1
= 0.00151
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No calculator, but I'm guessing the probability would be 50% just write down l te possibilities, I.e both people from 6 couple won, then 1 person from each of the couples, etc. Formula is impossible to write on iPod.