What is the Probability of getting a boy from 1000 families with 4 children each

Question

Out of 1000 families with 4 children each, how many would you expect to be at least a boy, given that the probbility of a female birth is 0.5

Answer 1

The possibilities are 4 boys, 3 boys and a girl, 2 boys and 2 girls, 3 girls and one boy, 4 girls.

This is a binomial distribution. The probability coefficients of these combinations is 1, 4, 6, 4, 1 respectively:

0.5^4, 4*0.5^4, 6*0.5^4, 4*0.5^4 and 0.5^4 because there is an equal chance of a boy or girl. (This can also be written: 1/16, 1/4, 3/8, 1/4, 1/16 which added together come to 1.)

The probability of 4 girls is (0.5)^4=0.0625 (1/16). Since all the other combinations contain at least one boy, the probability of at least one boy is 1-0.0625=0.9375 (15/16). In 1000 families with 4 children to get the expectation we multiply the probability by 1000=937.5. So we would expect 937 or 938 to contain at least one boy.