Question

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

Answer 1

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

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* 0.03125=0`

1
`p(1)={5\choose 1}(0.50)^(1)(1-0.50)^(5-1)=0.15625`
`1* 0.15625=0.15625`

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

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

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

5
`p(5)={5\choose 5}(0.50)^(5)(1-0.50)^(5-5)=0.03125`
`5* 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.