Chapter 16, Problem 5E

(a)

To determine

To construct: a probability model for the number of children they may have.

Explanation of Solution

Given:

mention that boys and girls are equality. Therefore, probability of having boy or girl is same and is $\frac{1}{2}$

They would not have more than 3 children.

They have children until they get a girl

Calculation:

Suppose X be number of children. Following is probability model of X

They will have 2 children if first is boy and second is girl

Suppose G shows girl and B shows Boy

  $\begin{array}{c}P\left(BG\right)=\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\\ =\frac{1}{4}\end{array}$

They stop with 3 even if 3^rd is boy or girl

  $\begin{array}{c}P\left(x=3\right)=P\left(BBG\right)+P\left(BBB\right)\\ =\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\\ =\frac{1}{8}+\frac{1}{8}\\ =\frac{1}{4}\end{array}$

The probability distribution is,

X	1(G)	2(BG)	3(BBG or BBB)
P(X= x)	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{4}$

(b)

To determine

To find: the expected number of children.

Answer to Problem 5E

1.75

Explanation of Solution

Formula used:

  $E\left(X\right)={\displaystyle \sum xP\left(X=x\right)}$

Calculation:

The expected number of children is

  $\begin{array}{c}E\left(X\right)={\displaystyle \sum xP\left(x\right)}\\ =1\left(\frac{1}{2}\right)+2\left(\frac{1}{4}\right)+3\left(\frac{1}{4}\right)\\ =\frac{1}{2}+\frac{2}{4}+\frac{3}{4}\\ =\frac{7}{4}\\ =1.75\end{array}$

Expected number of children is 1.75

(c)

To determine

To find: the expected number of boys they’ll have.

Answer to Problem 5E

0.875

Explanation of Solution

Formula used:

  $E\left(X\right)={\displaystyle \sum xP\left(X=x\right)}$

Calculation:

Probability model of X

X	0(G)	1 (BG)	2 (BBG)	3 (BBB)
P(X=x)	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$	$\frac{1}{8}$

  $\begin{array}{c}E\left(X\right)={\displaystyle \sum xP\left(x\right)}\\ =0\left(\frac{1}{2}\right)+1\left(\frac{1}{4}\right)+2\left(\frac{1}{8}\right)+3\left(\frac{1}{8}\right)\\ =\frac{1}{4}+\frac{2}{8}+\frac{3}{8}\\ =\frac{7}{8}\\ =0.875\end{array}$

Expected number of boys is 0.875