Expectation

Probability Level 2

Out of 4000 families with 5 children each, how many families would you expect to have at least 1 boy?

Note: Assume that the probability of a male birth is 1 2 \frac{1}{2} .


The answer is 3875.

A Former Brilliant Member by A Former Brilliant Member

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2 solutions

The probability that a family has at least one boy is the complement of the probability that a family includes no boys. The probability that a family has no boys is ( 1 / 2 ) 5 = 1 / 32 (1/2)^{5} = 1/32 , so the probability that a family has at least one boy is 1 − ( 1 / 32 ) = 31 / 32 1 - (1/32) = 31/32 , and so out of 4000 4000 families we would expect ( 31 / 32 ) × 4000 = 3875 (31/32) \times 4000 = \boxed{3875} families to have at least one boy.

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We could also calculate the complement, i.e., the number of families with no boys, and then subtract this value from 4000. The probability of there being no boys is ( 1 / 2 ) 5 = 1 / 32 (1/2)^{5} = 1/32 , which represents ( 1 / 32 ) × 4000 = 125 (1/32) \times 4000 = 125 families, and thus we would expect there to be 4000 − 125 = 3875 4000 - 125 = 3875 families with at least one boy.

Brian Charlesworth - 4 years, 2 months ago

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You should put that in as a solution so that others can see it more easily. (That's how I did it too)

Richard Costen - 4 years, 2 months ago

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O.k., done. :)

Brian Charlesworth - 4 years, 2 months ago

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