Chapter 0.6, Problem 17E

(a)

To determine

Probability for student is a freshman has not attended a game.

Answer to Problem 17E

Probability that student is a freshman has not attended a game is approx. 0.79.

Explanation of Solution

Given information:

Number of students who have attended a football game at North Coast High School:

Class	Freshman	Sophomore	Junior	Senior
attended	48	90	224	254
not attended	182	141	36	8

Calculations:

According to the conditional probability,

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

Calculate the table total:

Class	Freshman	Sophomore	Junior	Senior	Total
attended	48	90	224	254	616
not attended	182	141	36	8	367
Total	230	231	260	262	983

Note that

The information about 983 students is provided in the table.

Thus,

The number of possible outcomes is 983.

Also note that

In the table, 182 freshman students of the 983total students not attended the class.

Thus,

The number of favorable outcomes is 182.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

  $P\left(FreshmanandNotattended\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{182}{983}$

Now,

Note that

In the table, 230 of 983 students are freshman.

In this case, the number of favorable outcomes is 230 and number of possible outcomes is 983.

  $P\left(Freshman\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{230}{983}$

Apply the conditional probability:

  $\begin{array}{l}P\left(Notattended|Freshman\right)=\frac{P\left(FreshmanandNotattended\right)}{P\left(Freshman\right)}\\ =\frac{\frac{182}{983}}{\frac{230}{983}}\\ =\frac{182}{230}\\ =\frac{91}{115}\\ \approx 0.79\end{array}$

Thus,

The conditional probability for student is a freshman has not attended a game is approx. 0.79.

(b)

To determine

Probability for student has attended a game is an upperclassman (a junior or a senior).

Answer to Problem 17E

Probability that student has attended a game is an upperclassman is approx. 0.78.

Explanation of Solution

Given information:

Number of students who have attended a football game at North Coast High School:

Class	Freshman	Sophomore	Junior	Senior
attended	48	90	224	254
not attended	182	141	36	8

Calculations:

According to the conditional probability,

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

In this case, table becomes

Class	Freshman	Sophomore	Upperclassman	Total
attended	48	90	478	616
not attended	182	141	44	367
Total	230	231	522	983

Note that

The information about 983 students is provided in the table.

Thus,

The number of possible outcomes is 983.

Also note that

In the table calculated above, 478 of the 983 students attended the class are upperclassman.

Thus,

The number of favorable outcomes is 478.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

  $P\left(AttendedandUpperclassman\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{478}{983}$

Now,

Note that

In the table, 616 of 983 students attended the class.

In this case, the number of favorable outcomes is 616 and number of possible outcomes is 983.

  $P\left(Attended\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{616}{983}$

Apply the conditional probability:

  $\begin{array}{l}P\left(Upperclassman|Attended\right)=\frac{P\left(AttendedandUpperclassman\right)}{P\left(Attended\right)}\\ =\frac{\frac{478}{983}}{\frac{616}{983}}\\ =\frac{478}{616}\\ =\frac{239}{308}\\ \approx 0.78\end{array}$

Thus,

The conditional probability for student has attended a game is an upperclassman isapprox. 0.78.