Chapter 4.3, Problem 17E

(a)

To determine

To find:

The probability that Mrs. Harvey selects a male, given that she chooses from only the sophomores.

Answer to Problem 17E

Solution:

The required probability is 0.444.

Explanation of Solution

Given information:

Mrs. Harvey’s algebra class has 42 students, classified by academic year and gender as follows.

Mrs. Harvey’s Algebra Class
	Male	Female
Freshman	9	13
Sophomore	4	5
Junior	4	2
Senior	2	3

Definition:

Dependent Event:

Two events are dependent if one event happening effect the probability of the other event happening.

Formula used:

If $E\text{and}F$ are two dependent events, then the formula to calculate the probability that $F$ occurs given that $E$ occurs, is,

$P\left(F/E\right)=\frac{P\left(E\text{and}F\right)}{P\left(E\right)}$

Calculation:

Total number of student is 42.

There are 4 male and 5 female sophomores.

Total number of sophomore is 9.

Let $A$ be the event of choosing a male and $B$ be the event of choosing a sophomore.

Calculate the probability of event $A\text{and}B$.

$P\left(A\text{and}B\right)=\frac{4}{42}$

Calculate the probability of event $B$.

$P\left(B\right)=\frac{9}{42}$

Use the formula to calculate the conditional probability of event $A$, when another event $B$ has already occurred.

$\begin{array}{rll}P\left(A/B\right)& =& \frac{P\left(A\text{and}B\right)}{P\left(B\right)}\\ & =& \raisebox{1ex}{$\frac{4}{42}$}\!\left/ \!\raisebox{-1ex}{$\frac{9}{42}$}\right.\\ & =& \frac{4}{9}\\ & =& 0.444\end{array}$

Conclusion:

Thus, the probability that Mrs. Harvey selects a male, given that she chooses from only the sophomores is 0.444.

(b)

To determine

To find:

The probability that Mrs. Harvey selects a junior, given that she chooses a female.

Answer to Problem 17E

Solution:

The required probability is 0.0870.

Explanation of Solution

Given information:

Mrs. Harvey’s algebra class has 42 students, classified by academic year and gender as follows.

Mrs. Harvey’s Algebra Class
	Male	Female
Freshman	9	13
Sophomore	4	5
Junior	4	2
Senior	2	3

Definition:

Dependent Event:

Two events are dependent if one event happening effect the probability of the other event happening.

Formula used:

If $E\text{and}F$ are two dependent events, then the formula to calculate the probability that $F$ occurs given that $E$ occurs, is,

$P\left(F/E\right)=\frac{P\left(E\text{and}F\right)}{P\left(E\right)}$

Calculation:

Total number of student is 42.

There are 4 male and 2 female junior.

Total number of junior is 6.

Total number of female is $\left(13+5+2+3\right)=23$.

Let $A$ be the event of choosing a junior and $B$ be the event of choosing a female.

Calculate the probability of event $A\text{and}B$.

$P\left(A\text{and}B\right)=\frac{6}{42}$

Calculate the probability of event $B$.

$P\left(B\right)=\frac{23}{42}$

Use the formula to calculate the conditional probability of event $A$, when another event $B$ has already occurred.

$\begin{array}{rll}P\left(A/B\right)& =& \frac{P\left(A\text{and}B\right)}{P\left(B\right)}\\ & =& \raisebox{1ex}{$\frac{6}{42}$}\!\left/ \!\raisebox{-1ex}{$\frac{23}{42}$}\right.\\ & =& \frac{6}{23}\\ & =& 0.0870\end{array}$

Conclusion:

Thus, the probability that Mrs. Harvey selects a junior, given that she chooses a female is 0.0870.

(c)

To determine

To find:

The probability that Mrs. Harvey selects a female, given that she chooses a junior.

Answer to Problem 17E

Solution:

The required probability is 0.333.

Explanation of Solution

Given information:

Mrs. Harvey’s algebra class has 42 students, classified by academic year and gender as follows.

Mrs. Harvey’s Algebra Class
	Male	Female
Freshman	9	13
Sophomore	4	5
Junior	4	2
Senior	2	3

Definition:

Dependent Event:

Two events are dependent if one event happening effect the probability of the other event happening.

Formula used:

If $E\text{and}F$ are two dependent events, then the formula to calculate the probability that $F$ occurs given that $E$ occurs, is,

$P\left(F/E\right)=\frac{P\left(E\text{and}F\right)}{P\left(E\right)}$

Calculation:

Total number of student is 42.

There are 4 male and 2 female junior.

Total number of junior is 6.

Total number of female is $\left(13+5+2+3\right)=23$.

Let $A$ be the event of choosing a female and $B$ be the event of choosing a junior.

Calculate the probability of event $A\text{and}B$.

$P\left(A\text{and}B\right)=\frac{2}{42}$

Calculate the probability of event $B$.

$P\left(B\right)=\frac{6}{42}$

Use the formula to calculate the conditional probability of event $A$, when another event $B$ has already occurred.

$\begin{array}{rll}P\left(A/B\right)& =& \frac{P\left(A\text{and}B\right)}{P\left(B\right)}\\ & =& \raisebox{1ex}{$\frac{2}{42}$}\!\left/ \!\raisebox{-1ex}{$\frac{6}{42}$}\right.\\ & =& \frac{2}{6}\\ & =& 0.333\end{array}$

Conclusion:

Thus, the probability that Mrs. Harvey selects a female, given that she chooses a junior is 0.333.