Probability question about married couples winning a prize!
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Probability question about married couples winning a prize!

[From: ] [author: ] [Date: 11-09-22] [Hit: ]
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.-Imagine the people drawing tickets, one at a time,......
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.

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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

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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.
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