Question

Families with 5 children are randomly selected find the expected number of boys a family might have

Answer 1

Answer:

The expected number of boys a family might have is 2.5.

Step-by-step explanation:

Let X denote the number of boys in a family with 5 children.

The probability of a boy is, p = 0.50.

The child can either be a by or a girl independently.

The random variable X follows a binomial distribution with parameters n = 5 and p = 0.50.

The probability mass function is:

$p_{X}(x)={5\choose x}(0.50)^{x}(1-0.50)^{5-x};\ x=0,1,2...5$

Compute the expected number of boys a family might have as follows:

$E(X)=\sum x\cdot p_{X}(x)$

x                                      p (x)                                           x · p (x)

0       $p(0)={5\choose 0}(0.50)^{0}(1-0.50)^{5-0}=0.03125$          $0\times 0.03125=0$

1        $p(1)={5\choose 1}(0.50)^{1}(1-0.50)^{5-1}=0.15625$          $1\times 0.15625=0.15625$

2       $p(2)={5\choose 2}(0.50)^{2}(1-0.50)^{5-2}=0.3125$           $2\times 0.3125=0.625$

3       $p(3)={5\choose 3}(0.50)^{3}(1-0.50)^{5-3}=0.3125$           $3\times 0.3125=0.9375$

4       $p(4)={5\choose 4}(0.50)^{4}(1-0.50)^{5-4}=0.15625$         $4\times 0.15625=0.625$

5       $p(5)={5\choose 5}(0.50)^{5}(1-0.50)^{5-5}=0.03125$         $5\times 0.03125=0.15625$

Then,

$E(X)=\sum x\cdot p_{X}(x)$

         $=0+0.15625+0.625+0.9375+0.625+0.15625\\\\=2.5$

Thus, the expected number of boys a family might have is 2.5.