Question

Out of 750 families with 4 children each, how many families would be expected to have children of both sexes? Assume equal probabilities for boys and girls.

Answer 1

Assume equal probabilities for boys and girls

Let p be the probability of having a boy

Let x be the random variable for getting either a boy or a girl

∴ p = `1/2` and q = `1/2` and n = 4

In binomial distribution P(X = 4) = nc_xp^xq^n-x

Here the binomial distribution is P(X= x) = `4"C"_x (1/2)^x (1/2)^("n" - x)`

P(children of both sexes) = P(X = 1) + P(X = 2) + P(X = 3)

= `4"C"_1 (1/2)^1 (1/2)^(4 - 2) + 4"C"_2 (1/2)^2 (1/2)^(4 - 2) + 4"C"_3 (1/2)^3 (1/2)^(4 - 3)`

= `4 xx (1/16) + 6 xx (1/16) + 4 xx 1/16`

= `1/16[4 + 6 + 4]`

= `14/16`

= 0.875 x 750

For 750 families P(X = 2) = 0.875 × 750

= 656.25

= 656 ......(approximately)