SOLUTION: Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them with a certain trait is 2/5. If they have five children, what is the probability

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Question 1208629: Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them with a certain trait is 2/5. If they have five children, what is the probability that at least two of their five children will have that trait? Round your answer to the nearest thousandth.
Answer by Shin123(626) About Me  (Show Source):
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We will use complementary counting. This means we want to find the probability that 0 or 1 of their children have the trait.
The probability of none of their children having the trait is %283%2F5%29%5E5, since the probability for an individual child is 1-2%2F5=3%2F5, and there are 5 of them.
The probability of exactly 1 of their children having the trait is 5%2A%282%2F5%29%2A%283%2F5%29%5E4. The probability of a child having the trait is 2%2F5, and the probability of the other 4 not having the trait is %281-2%2F5%29%5E4=%283%2F5%29%5E4. The child with the trait could be any of the 5, so we multiply by 5 to account for that.
So in total, the probability of 0 or 1 of their children having the trait is %283%2F5%29%5E5%2B5%2A%282%2F5%29%2A%283%2F5%29%5E4, or 0.33696. Since that is the probability of there not being at least 2 children having the trait, the answer is 1-0.33696=0.66304, which rounds to 0.663.