Chapter 3.2, Problem 27AQP

a.

Summary Introduction

To determine:

The probability of obtaining all boys in a family of seven children.

Explanation of Solution

There are total seven children in the family. The probability of obtaining all boys can be calculated by multiplication rule. The probability of a single boy is half. Therefore, the probability of obtaining seven children as boys can be calculated as:

$\begin{array}{c}\text{Probability}\text{of obtatining}\text{a}\text{single}\text{boy}\text{=}\frac{1}{2}\\ \text{Probability}\text{of obtatining}\text{all}\text{seven}\text{boys}={\left(\frac{1}{2}\right)}^7=\frac{1}{128}\end{array}$

Hence the probability of obtaining all seven children as boys is $\frac{1}{128}$.

b.

Summary Introduction

To determine:

The probability of obtaining all children of the same sex in a family of seven children.

Explanation of Solution

There are total seven children in the family. The probability of a single boy and single girl is half. Therefore, the probability of obtaining all children of same sex can be calculated as:

$\begin{array}{c}\text{Probability}\text{of obtatining}\text{a}\text{single}\text{boy}\text{=}\frac{1}{2}\\ \text{Probability}\text{of obtatining}\text{all}\text{boys}={\left(\frac{1}{2}\right)}^7=\frac{1}{128}\end{array}$

$\begin{array}{c}\text{Probability}\text{of obtatining}\text{a}\text{single girl}\text{=}\frac{1}{2}\\ \text{Probability}\text{of obtatining}\text{all girls}={\left(\frac{1}{2}\right)}^7=\frac{1}{128}\end{array}$

The probability of obtaining all children of same sex can be calculated by addition rule as:

$\begin{array}{c}\text{Probability}\text{of obtatining}\text{either}\text{all}\text{girls}\text{or}\text{all}\text{boys}\text{=}\text{Probability}\text{of obtatining}\text{all boys +}\\ \text{ Probability}\text{of obtatining}\text{all}\text{girls}\\ =\frac{1}{128}+\frac{1}{128}\\ \text{Probability}\text{of obtatining}\text{either}\text{all}\text{girls}\text{or}\text{all}\text{boys}=\frac{1}{64}\end{array}$

Hence the probability of obtaining all children of same sex is $\frac{1}{64}$.

c.

Summary Introduction

To determine:

The probability of obtaining six girls and one boy in a family of seven children.

Explanation of Solution

The binomial expansion ${\left(\text{x+y}\right)}^{\text{n}}$ is used to calculate the probabilities of events where x and y represents the respective events and n is the total number of times an event occurs.

In the given case x is the probability of obtaining a girl and y is the probability of obtaining a boy. n is seven as there are total seven children in the family. The binomial expansion for this case will be given as:

${\left(\text{x+y}\right)}^{\text{n}}\text{=}{\text{x}}^{\text{7}}{\text{+7x}}^{\text{6}}\text{y}{\text{+21x}}^{\text{5}}{\text{y}}^{\text{2}}{\text{+35x}}^{\text{4}}{\text{y}}^{\text{3}}{\text{+35x}}^3{\text{y}}^4{\text{+21x}}^{\text{2}}{\text{y}}^{\text{5}}\text{+}{\text{7xy}}^{\text{6}}{\text{+y}}^7$

The probability of obtaining six girls and one boy is expressed by ${\text{7x}}^{\text{6}}\text{y}$ binomial term as x is the probability of obtaining girls and y is the probability of obtaining boys. The probability of a single boy and single girl is half. By using ${\text{7x}}^{\text{6}}\text{y}$ binomial term, the probability of obtaining six girls and one boy can be calculated as:

$\begin{array}{c}\text{The probability of obtaining six girls and one boy}=7{\left(\frac{1}{2}\right)}^6\left(\frac{1}{2}\right)\\ =\frac{7}{128}\end{array}$

Thus the probability of obtaining six girls and one boy is $\frac{7}{128}$.

d.

Summary Introduction

To determine:

The probability of obtaining four boys and three girls in a family of seven children.

Explanation of Solution

The binomial expansion for the family of seven children will be given as:

${\left(\text{x+y}\right)}^{\text{n}}\text{=}{\text{x}}^{\text{7}}{\text{+7x}}^{\text{6}}\text{y}{\text{+21x}}^{\text{5}}{\text{y}}^{\text{2}}{\text{+35x}}^{\text{4}}{\text{y}}^{\text{3}}{\text{+35x}}^3{\text{y}}^4{\text{+21x}}^{\text{2}}{\text{y}}^{\text{5}}\text{+}{\text{7xy}}^{\text{6}}{\text{+y}}^7$

The probability of obtaining three girls and four boys is expressed by ${\text{35x}}^3{\text{y}}^4$ binomial term as x is the probability of obtaining girls and y is the probability of obtaining boys. The probability of a single boy and single girl is half. By using ${\text{35x}}^3{\text{y}}^4$ binomial term, the probability of obtaining three girls and four boys can be calculated as:

$\begin{array}{c}\text{The probability of obtaining three girls and four boys}=35{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^4\\ =\frac{35}{128}\end{array}$

Thus, the probability of obtaining three girls and four boys is $\frac{35}{128}$.

e.

Summary Introduction

To determine:

The probability of obtaining four girls and three boys in a family of seven children.

Explanation of Solution

The binomial expansion for the family of seven children will be given as:

${\left(\text{x+y}\right)}^{\text{n}}\text{=}{\text{x}}^{\text{7}}{\text{+7x}}^{\text{6}}\text{y}{\text{+21x}}^{\text{5}}{\text{y}}^{\text{2}}{\text{+35x}}^{\text{4}}{\text{y}}^{\text{3}}{\text{+35x}}^3{\text{y}}^4{\text{+21x}}^{\text{2}}{\text{y}}^{\text{5}}\text{+}{\text{7xy}}^{\text{6}}{\text{+y}}^7$

The probability of obtaining four girls and three boys is expressed by ${\text{35x}}^{\text{4}}{\text{y}}^{\text{3}}$ binomial term as x is the probability of obtaining girls and y is the probability of obtaining boys. The probability of a single boy and single girl is half. By using ${\text{35x}}^{\text{4}}{\text{y}}^{\text{3}}$ binomial term, the probability of obtaining four girls and three boys can be calculated as:

$\begin{array}{c}\text{The probability of obtaining four girls and three boys}=35{\left(\frac{1}{2}\right)}^4{\left(\frac{1}{2}\right)}^3\\ =\frac{35}{128}\end{array}$

Thus, the probability of obtaining four girls and three boys is $\frac{35}{128}$.

Conclusion

The probability of all boys is $\frac{1}{128}$, of obtaining same sex children is $\frac{1}{64}$, of six girls and one boy is $\frac{7}{128}$, of three girls and four boys is $\frac{35}{128}$ and obtaining four girls and three boys is $\frac{35}{128}$ in a family of seven children.