College Algebra, Chapter 10, 10.4, Section 10.4, Problem 22

Assume that for any given live human birth, the chances that a child is a boy or girl are equally likely.

a.) What is the probability that in a family of five children a majority are boys?

b.) What is the probability that in a family of seven children a majority are girls?

Recall that the formula for the binomial probability is given by

$C(n,r) p^r q^{n-r}$

In this case, the probability of success and failure is $p = q = 0.50$.

a.) If the family of five children has majority of boys, then the number of boys must be 3. In this case, $r=3$. So, we have

$= C(5,3)(0.50)^3 (0.50)^{5-3}$

$= C(5,3)(0.50)^3 (0.50)^2$

$= 0.3125$

b.) If the family of seven children has majority of girls, then the number of girls must be 4. In this case, $r=4$, so

$= C(7,4) (0.50)^4 (0.50)^{7-3}$

$= C(7,4) (0.50)^4 (0.50)^3 $

$= 0.2734$