Chapter 12, Problem 12.48E

a.

To determine

To write: All the eight possible arrangements for the boy and girl child.

To find: The probability for any one of the eight possible arrangements for the child.

Answer to Problem 12.48E

The eight possible arrangements for the boy and girl child are given below,

$S=\left\{\begin{array}{l}\left(GGB\right),\left(GBG\right),\left(BGG\right),\left(GGG\right)\\ \left(BGG\right),\left(BGB\right),\left(GBB\right),\left(BBB\right)\end{array}\right\}$

The probability to get first two girls and a boy is $\frac{1}{8}$ and the probability is the same for other possible arrangements.

Explanation of Solution

Given info:

A couple has planned to have three child, eight possible arrangements for the girl and boy are possible. All these eight arrangements are equally likely to occurrence.

Justification:

Sample space:

The sample space is defined as the set of all possible outcomes from an experiment.

The total number of possible outcomes is,

$\begin{array}{c}\text{Total number of possible outcomes}={\left(\text{Possible outcomes for a child}\right)}^{\left(\text{Number of children}\right)}\\ =2^3\\ =8\end{array}$

A couple plans to have three children. One of the possible combinations to have three children is “Girl, Girl or Boy”. Similarly, the other possible combinations can be obtained.

All the eight possible arrangements for the boy and girl child are the sample space S which is given below:

$S=\left\{\begin{array}{l}\left(GGB\right),\left(GBG\right),\left(BGG\right),\left(GGG\right)\\ \left(BGG\right),\left(BGB\right),\left(GBB\right),\left(BBB\right)\end{array}\right\}$

Where, G represents the girl child and B represents the boy child.

Calculation:

Equally likely events:

An event is said to be equally likely if all the possible outcomes has equal chance of occurrence.

Since, the eight possible arrangements are equally likely to occur. The probability for one of the eight possible arrangements is calculated as follows:

$\begin{array}{c}P\left(\begin{array}{l}\text{Any one of the eight}\\ \text{possible arrangements}\end{array}\right)=P\left(\frac{\begin{array}{l}\text{Number of ways in which any one of the eight}\\ \text{possible arrangements could occur}\end{array}}{\text{Total number of possible arrangements}}\right)\\ =\frac{1}{8}\end{array}$

Thus, the probability for getting any one from the eight possible arrangements is $\frac{1}{8}$ and the probability is the same for other possible arrangements.

b.

To determine

To find: The probability that $X=2$.

Answer to Problem 12.48E

The probability that the couple have two girl children is $\frac{4}{8}$.

Explanation of Solution

Given info:

Assume that X denotes the number of girls that the couple has.

Calculation:

Let the number of girls X that the couple has equals to 2 girls.

The outcomes for 2 girls are $\left\{\left(GGB\right),\left(GBG\right),\left(BGG\right),\left(BGG\right)\right\}$

The probability that the couple have two girl children is calculated as follows:

$\begin{array}{c}P\left(\begin{array}{l}\text{Couple having two}\\ \text{girl children}\end{array}\right)=\frac{\text{Number of ways in which the couple could have two girls}}{\text{Total number of ways to have three children}}\\ =\frac{4}{8}\end{array}$

Thus, the probability that the couple have two girl children is $\frac{4}{8}$.

c.

To determine

To find: The values of X and the probability distribution for X.

Answer to Problem 12.48E

The values taken by X are 0, 1, 2, and 3.

The probability distribution is given below:

X	0	1	2	3
Probability	$\frac{1}{8}$	$\frac{2}{8}$	$\frac{4}{8}$	$\frac{1}{8}$

Explanation of Solution

Calculation:

Random variable:

The random variable is a variable which has numerical values or outcomes obtained from a random experiment.

Finite Probability Model:

A probability model with a finite sample space is called the finite probability model.

Assigning probabilities to the finite probability model:

• List all the probabilities for all individual outcomes.
• These probabilities should lie between 0 and 1 and the total sum of all probabilities exactly equal to 1.
• The probability for occurrence of any event is the sum of individual probabilities of that event.

Values of X:

The random variable X takes values 0, 1, 2, 3 because the couple has planned to have 3 children and X denotes the number of girl child. So, the possible number of girl child the couple can have is 0, 1, 2, and 3.

Probability distribution for X:

Let the number of girls X that the couple has equals to 0. The possible outcome is $\left\{\left(BBB\right)\right\}$

The probability to have no girl child is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Probability to have}\\ \text{no girl child}\end{array}\right)=\frac{\text{Number of ways in which no girl child can born}}{\text{Number of ways to have three children}}\\ =\frac{1}{8}\end{array}$

Thus, the probability to have no girl child is $\frac{1}{8}$

Let the number of girls X that the couple has equals 1. The possible outcomes are $\left\{\left(BGB\right),\left(GBB\right)\right\}$

The probability to have 1 girl child is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Probability to have}\\ \text{1 girl child}\end{array}\right)=\frac{\text{Number of ways in which one girl child can born}}{\text{Number of ways to have three children}}\\ =\frac{2}{8}\end{array}$

Thus, the probability to have one girl child is $\frac{2}{8}$

Let the number of girls X that the couple has equals 2. The outcomes for 2 girls are $\left\{\left(GGB\right),\left(GBG\right),\left(BGG\right),\left(BGG\right)\right\}$

The probability to have 2 girl children is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Probability to have}\\ \text{2 girl child}\end{array}\right)=\frac{\text{Number of ways in which two girl child can born}}{\text{Number of ways to have three children}}\\ =\frac{4}{8}\end{array}$

Thus, the probability to have two girl child is $\frac{4}{8}$.

Let the number of girls X that the couple has equals 3. The possible outcome is $\left\{\left(GGG\right)\right\}$

The probability to have 3 girl children is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}\text{Probability to have}\\ \text{3 girl child}\end{array}\right)=\frac{\text{Number of ways in which three girl child can born}}{\text{Number of ways to have three children}}\\ =\frac{1}{8}\end{array}$

Thus, the probability to have three girl child is $\frac{1}{8}$.

The probability distribution is given below:

X	0	1	2	3
Probability	$\frac{1}{8}$	$\frac{2}{8}$	$\frac{4}{8}$	$\frac{1}{8}$