Chapter 5.3, Problem 43E

Stay in school: In a recent school year in the state of Washington, there were 326,000 high school students. Of these, 159,000 were girls and 167,000 were boys. Among the girls, 7800 dropped out of school, and among the boys, 10,300 dropped out. A student is chosen at random.

1. What is the probability that the student is male?
2. What is the probability that the student dropped out?
3. What is the probability that the student is male arid dropped out?
4. Given that the student is male, what is die probability that he dropped out?
5. Given that the student dropped out, what is the probability that the student is male?

To determine

The probability that randomly chosen student is male.

Answer to Problem 43E

  $0.5123$

Explanation of Solution

Given:

Total number of students is 326,000.Total number of girls is 159,000 and boys is 167,000.

Number of droppedout from girls is 7800 and that of boys is 10,300.

Formula Used:

Probability of an event is given by

  $\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let $M$ be the event that selectedstudent is male.

  $\begin{array}{c}n\left(M\right)=167,000\\ n\left(S\right)=326,000\\ P\left(M\right)=\frac{167,000}{326,000}\\ =0.5123\end{array}$

Hence, the probability that randomly chosen student is male,is $0.5123$ .

To determine

The probability that randomly chosen student is dropped out.

Answer to Problem 43E

  $0.555$

Explanation of Solution

Given:

Total number of students is 326,000.Total number of girls is 159,000 and boys is 167,000.

Number of droppedout from girls is 7800 and that of boys is 10,300.

Formula Used:

Probability of an event is given by

  $\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let $D$ be the event that selected student is male.

  $\begin{array}{c}n\left(D\right)=10300+7800=18100\\ n\left(S\right)=326,000\\ P\left(D\right)=\frac{18100}{326,000}\\ =0.555\end{array}$

Hence, the probability that randomly chosen student is dropped out, is $0.555$ .

To determine

The probability that the student is dropped outand male.

Answer to Problem 43E

  $0.0316$

Explanation of Solution

Given:

Total number of students is 326,000.Total number of girls is 159,000 and boys is 167,000.

Number of droppedout from girls is 7800 and that of boys is 10,300.

Formula Used:

For any two t events $A$ and B , probability of occurring both the events is

  $\begin{array}{c}P\left(A\text{and}B\right)=P\left(A\cap \text{}B\right)\\ =\frac{n\left(A\cap \text{}B\right)}{n\left(S\right)}\end{array}$

Calculation:

Let $M$ be the event that selected student is male.

Let $D$ be the event that selected student is male.

So, $M\text{and}D$ be the event that selected student is dropped out and male.

There are 10,3000male who are dropped out.

  $\begin{array}{c}P\left(M\cap D\right)=\frac{n\left(M\cap D\right)}{n\left(S\right)}\\ =\frac{10,300}{326,000}\\ =0.0316\end{array}$

The probability that the thestudent is dropped out and male, is $0.0316$ .

To determine

The probability to a student of being dropped out, given that student is male.

Answer to Problem 43E

  $0.0617$

Explanation of Solution

Given:

Total number of students is 326,000.Total number of girls is 159,000 and boys is 167,000.

Number of droppedout from girls is 7800 and that of boys is 10,300.

Formula Used:

For any two dependent events $A$ and B ,

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

Where

  $P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let $M$ be the event that selected student is male.

  $\begin{array}{c}n\left(M\right)=167,000\\ n\left(S\right)=326,000\\ P\left(M\right)=\frac{167,000}{326,000}\\ =0.5123\end{array}$

Let $D$ be the event that selected student is male.

So, $M\text{and}D$ be the event that selected student is dropped out and male.

There are 10,3000male who are dropped out.

  $\begin{array}{c}P\left(M\cap D\right)=\frac{n\left(M\cap D\right)}{n\left(S\right)}\\ =\frac{10,300}{326,000}\\ =0.0316\end{array}$

Therefore, $D/M$ be the event that a student of being dropped out, given that student is male.

So, the probability of the event “a student of being dropped out, given that student is male” is calculated as follows:

  $\begin{array}{c}P\left(D/M\right)=\frac{P\left(M\text{and}D\right)}{P\left(M\right)}\\ =\frac{0.0316}{0.5123}\\ =0.0617\end{array}$

The probability to a student of being dropped out, given that student is male is $0.0617$ .

To determine

The probability to a student of being male, given that student is dropped out.

Answer to Problem 43E

  $0.5691$

Explanation of Solution

Given:

Total number of students is 326,000.Total number of girls is 159,000 and boys is 167,000.

Number of droppedout from girls is 7800 and that of boys is 10,300.

Formula Used:

For any two dependent events $A$ and B ,

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

Where

  $P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let $D$ be the event that selected student is male.

  $\begin{array}{c}n\left(D\right)=10300+7800=18100\\ n\left(S\right)=326,000\\ P\left(D\right)=\frac{18100}{326,000}\\ =0.555\end{array}$

Let $M$ be the event that selected student is male.

So, $M\text{and}D$ be the event that selected student is dropped out and male.

There are 10,3000male who are dropped out.

  $\begin{array}{c}P\left(M\cap D\right)=\frac{n\left(M\cap D\right)}{n\left(S\right)}\\ =\frac{10,300}{326,000}\\ =0.0316\end{array}$

Therefore, $M/D$ be the event that a student of being dropped out, given that student is male.

So, the probability of the event “a student of being dropped out, given that student is male” is calculated as follows:

  $\begin{array}{c}P\left(M/D\right)=\frac{P\left(M\text{and}D\right)}{P\left(D\right)}\\ =\frac{0.0316}{0.555}\\ =0.5691\end{array}$

The probability to a student of being male, given that student is dropped out is $0.5691$ .