SOLUTION: a family has 6 children, find the probability that fewer boys and than girls. the probability that any child being a boy is equal to half

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Question 1102441: a family has 6 children, find the probability that fewer boys and than girls. the probability that any child being a boy is equal to half
Answer by greenestamps(13203) About Me  (Show Source):
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The probability of having 0 boys is C%286%2C0%29%2A%281%2F2%29%5E6+=+%281%29%281%2F64%29+=+1%2F64.
The probability of having 1 boy is C%286%2C1%29%2A%281%2F2%29%5E6+=+%286%29%281%2F64%29+=+6%2F64.
The probability of having 2 boys is C%286%2C2%29%2A%281%2F2%29%5E6+=+%2815%29%281%2F64%29+=+15%2F64.

So the probability of having fewer boys than girls in a family with 6 children is 22/64 = 11/32.

For probability problems like this where there are two outcomes each with probability 1/2 (e.g., gender of children; flipping a fair coin), the easiest and fastest way to get to the answer is using Pascal's Triangle.

The 6th row of Pascal's Triangle is
1 6 15 20 15 6 1
and the sum of the entries is 2^6=64

A quick look at that row tells you the probability is...
1/64 of having 0 boys
6/64 of having 1 boy
15/64 of having 2 boys
20/64 of having 3 boys
15/64 of having 4 boys
6/64 of having 5 boys
1/64 of having 6 boys