Chapter 4.3, Problem 19E

Student Financial Aid In a recent year 8,073,000 male students and 10,980,000 female students were enrolled as undergraduates. Receiving aid were 60.6% of the male students and 65.2% of the female students. Of those receiving aid, 44.8% of the males got federal aid and 50.4% of the females got federal aid. Choose 1 student at random. (Hint: Make a tree diagram.) Find the probability that the student is

a. A male student without aid

b. A male student, given that the student has aid

c. A female student or a student who receives federal aid

Source: www.nces.gov

To determine

To find: The probability that the student is a male student without aid.

Answer to Problem 19E

The probability that the student is a male student without aid is 0.167.

Explanation of Solution

Given info:

There is one student is chosen at random. There are 8,073,000 male students and 10,980,000 female students in which aid were 60.6% of the male students and 65.2% of the female students. Of those receiving aid, 44.8% of the males got federal aid and 50.4% of the females got federal aid.

Calculation:

Multiplication rule for dependent events:

If the events A and B are dependent, the probability of occurring of event B is affected by event A, that is $P\left(A\text{ and }B\right)$ is $P\left(A\right)\times P\left(B|A\right)$.

Let event M denote that student is a male and WOA be denote that selecting student has without aids.

Here, there are 8,073,000 male students and 10,980,000 female students.

Also, the tree diagram is shown below:

The frequency for the class is 8,073,000. Also, the total frequency is,

$\begin{array}{c}n=8,073,000+10,980,000\\ =19,053,000\end{array}$

The formula for probability of event M is,

$P\left(M\right)=\frac{\text{Frequency for the class}}{\text{Total frequencies in the distribution}}$

Substitute 8,073,000 for ‘Frequency for the class’ and 19,053,000 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(M\right)=\frac{8,073,000}{19,053,000}\\ =0.424\end{array}$

Thus, the probability that the student is a male is 0.424.

Also, from the tree diagram it is observed that the probability that the without aid in male students are 0.394.

By applying multiplication rule, the probability that the student is a male student without aid is,

$\begin{array}{c}P\left(M\text{ and }WOA\right)=P\left(M\right)\times P\left(WOA|M\right)\\ =0.424\times 0.394\\ =0.167\end{array}$

Thus, the probability that the student is a male student without aid is 0.167.

To determine

To find: The probability that the student is a male student, given that the student has aid.

Answer to Problem 19E

The probability that the student is a male student, given that the student has aid is 0.406.

Explanation of Solution

Calculation:

Let event F denote that student is a female and WA be denote that selecting student has with aids.

From the part (a), it is observed that the probability that the student is a male student is 0.424then the probability that the student is a female student is 0.576 $\left(=1-0.424\right)$.

From, tree diagram it is observed that the probability that the students has aid in males is 0.606 and the probability that the students has aid in females is 0.652.

By applying multiplication rule, the probability that that the student has aid is,

The required probability is,

$\begin{array}{c}P\left(WA\right)=\left\{\left[P\left(M\right)\times P\left(WA|M\right)\right]+\left[P\left(F\right)\times P\left(WA|F\right)\right]\right\}\\ =\left\{\left[0.424\times 0.606\right]+\left[0.576\times 0.652\right]\right\}\\ =0.257+0.376\\ =0.633\end{array}$

Then, the probability that the student is a male student and the student has aid is,

$\begin{array}{c}P\left(M\text{ and }WA\right)=P\left(M\right)\times P\left(WA|M\right)\\ =0.424\times 0.606\\ =0.257\end{array}$

Then, the probability that the student is a male student, given that the student has aid is,

$\begin{array}{c}P\left(M|WA\right)=\frac{\left[P\left(M\right)\times P\left(WA|M\right)\right]}{\left\{\left[P\left(M\right)\times P\left(WA|M\right)\right]+\left[P\left(F\right)\times P\left(WA|F\right)\right]\right\}}\\ =\frac{0.257}{0.633}\\ =0.406\end{array}$

Thus, the probability that the student is a male student, given that the student has aid is 0.406.

To determine

To find: The probability that the student is a female student or a student who receives federal aid.

Answer to Problem 19E

The probability that the student is a female student or a student who receives federal aid is 0.691.

Explanation of Solution

Calculation:

Let event FA denote that student has federal aid.

From, part (b) it is observed that the student is a female student is 0.576. Also, from tree graph it is observed that the probability that the student has aid in males is 0.606 and in which federal aids are 0.448. Also, the probability that the student has aid in females is 0.652 and in which federal aids are 0.448.

By applying multiplication rule, the probability that the student has federal aid is,

$\begin{array}{c}P\left(FA\right)=\left\{\left[P\left(M\right)\times P\left(WA|M\right)\times P\left(FA|M\right)\right]+\left[P\left(F\right)\times P\left(WA|F\right)\times P\left(FA|F\right)\right]\right\}\\ =\left\{\left[0.424\times 0.606\times 0.448\right]+\left[0.576\times 0.652\times 0.448\right]\right\}\\ =0.115+0.168\\ =0.283\end{array}$

Then, the probability that the student is a female student and the student has federal aid is,

$\begin{array}{c}P\left(F\text{ and }FA\right)=\left[P\left(F\right)\times P\left(WA|F\right)\times P\left(FA|F\right)\right]\\ =\left[0.576\times 0.652\times 0.448\right]\\ =0.168\end{array}$

Addition Rule:

The formula for probability of getting event F or event FA is,

$P\left(F\text{ or }FA\right)=P\left(F\right)+P\left(FA\right)-P\left(F\text{ and }FA\right)$

Substitute 0.576 for $P\left(F\right)$, 0.283 for $P\left(FA\right)$ and 0.168 for $P\left(F\text{ and }FA\right)$

$\begin{array}{c}P\left(F\text{ or }FA\right)=0.576+0.283-0.168\\ =0.691\end{array}$

Thus, the probability that the student is a female student or a student who receives federal aid is 0.691