Suppose 10 couples attend an event,and all 20 people enter a raffle.Four of the 20 peopleare selected at random to win $10 in the raffle.What is the probability that at least onecouple takes home $20,which will occur if both the husband and wife win

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Suppose 10 couples attend an event,and all 20 people enter a raffle.Four of the 20 peopleare selected at random to win $10 in the raffle.What is the probability that at least onecouple takes home $20,which will occur if both the husband and wife win
Suppose 10 couples attend an event,and all 20 people enter a raffle.Four of the 20 peopleare selected at random to win $10 in the raffle.What is the probability that at least onecouple takes home $20,which will occur if both the husband and wife win the raffle?

Suppose 10 couples attend an event,and all 20 people enter a raffle.Four of the 20 peopleare selected at random to win $10 in the raffle.What is the probability that at least onecouple takes home $20,which will occur if both the husband and wife win
4个奖券总共有 C(20,4) 种分法.
要求至少有一个家庭中2份奖.
(1) 有2个家庭中2份奖:C(10,2)
(2) 有且仅有1个家庭中2份奖:C(10,1) * C(9,2) * C(2,1) * C(2,1)
先从10个家庭选出1个中2份奖的:C(10,2)
再从剩下9个家庭选出2个中1份奖的:C(9,2)
再从2个中1份奖的家庭中,每个家庭选出1个中奖的:C(2,1) * C(2,1)
综上,至少有1个家庭中2份奖的概率是:
(C(10,2) + C(10,1) * C(9,2) * C(2,1) * C(2,1)) / C(20,4)
= 0.30650154798761609907120743034056
或者这么算:
没有一个家庭中2份奖:C(10,4) * 2^4
所以,没有一个家庭中2份奖的概率:C(10,4) * 2^4 / C(20,4) = 0.693498452012384
所以,我们所求的是:1 - 0.693498452012384 = 0.306501547987616

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