Question

out of 300 families with 5 children each, what percentage would be expected to have

i. at least a boy

ii. all boys

iii. at most 2 girls

Answer 1

Let binomial random variable $X$ denotes the number of boys in the family: $X\sim B(n, p).$

Since boy and girl child have equal probability, then $p=q=0.5.$

Given $n=5.$ Then $X\sim B(5, 0.5)$

i.

$$P(X\geq1)=1-P(X<1)=1-P(X=0)$$

$$=1-\dbinom{5}{0}(0.5)^0(0.5)^{5-0}=0.96875$$

$96.875\ \%$

ii.

$$P(X=5)=\dbinom{5}{0}(0.5)^5(0.5)^{5-5}=0.03125$$

$3.125\ \%$

iii.

$$P(X\geq3)=P(X=3)+P(X=4)+P(X=5)$$

$$=\dbinom{5}{3}(0.5)^3(0.5)^{5-3}+\dbinom{5}{4}(0.5)^4(0.5)^{5-4}$$

$$+\dbinom{5}{5}(0.5)^5(0.5)^{5-5}=(10+5+1)(0.5)^5=0.5$$

$50\ \%$