what is the probability of eight boys in ten births

Question

Assuming that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children.  Find the probability of exactly eight boys in ten births.

Answer 1

Since the probabilities of male and female births are the same, p=½ and the probability of 8 boys out of 10 is the same as 2 girls out of 10.

(p(boy)+p(girl))¹⁰=p¹⁰+10p⁹p+45p⁸p²+...=p¹⁰+10p¹⁰+45p¹⁰+...

We need no more terms than these three which correspond to the probabilities:

All boys, 9 boys + 1 girl, 8 boys + 2 girls, ...

So we simply add the coefficients: 1+10+45=56, giving us 56×½¹⁰=56/1024=14/256=0.0547 approx or 5.47%.