Chapter 0.6, Problem 16E

(a)

To determine

Probability for club member is a male student.

Answer to Problem 16E

Probability that club member is a male student is approx. 0.39.

Explanation of Solution

Given information:

King high school data for number of males and females that were members of at least one after school club:

Gender	Clubs	No Clubs
male	156	242
female	312	108

Calculation:

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:

Status	Club	No Club	Total
male	156	242	398
female	312	108	420
Total	468	350	818

Note that

The information about 818 students is provided in the table.

Thus,

The number of possible outcomes is 818.

Also note that

In the table, 156 of the 818 students are male club members.

Thus,

The number of favorable outcomes is 156.

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

  $P\left(\text{Club}\text{member}\text{and}\text{male}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{156}{818}$

Now,

Note that

In the table, 398 of 818 students are male.

In this case, the number of favorable outcomes is 398 and number of possible outcomes is 818.

  $P\left(\text{male}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{398}{818}$

Apply the conditional probability:

  $\begin{array}{l}P\left(\text{club}\text{member}|\text{male}\right)=\frac{P\left(\text{club}\text{member}\text{and}\text{male}\right)}{P\left(\text{male}\right)}\\ =\frac{\frac{156}{818}}{\frac{398}{818}}\\ =\frac{156}{398}\\ =\frac{78}{199}\\ \approx 0.39\end{array}$

Thus,

The conditional probability for club member is a male student is approx. 0.39.

(b)

To determine

Probability for non − club member is a female student.

Answer to Problem 16E

Probability that non − club member is a female student is approx. 0.26.

Explanation of Solution

Given information:

King high school data for number of males and females that were members of at least one after school club:

Gender	Clubs	No Clubs
male	156	242
female	312	108

Calculation:

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:

Status	Club	No Club	Total
male	156	242	398
female	312	108	420
Total	468	350	818

Note that

The information about 818 students is provided in the table.

Thus,

The number of possible outcomes is 818.

Also note that

In the table, 108 of the 818 students are female non − club members.

Thus,

The number of favorable outcomes is 108.

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

  $P\left(\text{No}\text{Club}\text{member}\text{and}\text{female}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{108}{818}$

Now,

Note that

In the table, 420 of 818 students are female.

In this case, the number of favorable outcomes is 398 and number of possible outcomes is 818.

  $P\left(\text{female}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{420}{818}$

Apply the conditional probability:

  $\begin{array}{l}P\left(\text{no}\text{club}\text{member}|\text{female}\right)=\frac{P\left(\text{no}\text{club}\text{member}\text{and}\text{female}\right)}{P\left(\text{female}\right)}\\ =\frac{\frac{108}{818}}{\frac{420}{818}}\\ =\frac{108}{420}\\ =\frac{9}{35}\\ \approx 0.26\end{array}$

Thus,

The conditional probability for non − club member is a female student is approx. 0.26.

(c)

To determine

Probability for male student is not a club member.

Answer to Problem 16E

Probability that male student is not a club member is approx. 0.69.

Explanation of Solution

Given information:

King high school data for number of males and females that were members of at least one after school club:

Gender	Clubs	No Clubs
male	156	242
female	312	108

Calculation:

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:

Status	Club	No Club	Total
male	156	242	398
female	312	108	420
Total	468	350	818

Note that

The information about 818 students is provided in the table.

Thus,

The number of possible outcomes is 818.

Also note that

In the table, 242 of the 818 students are male non − club members.

Thus,

The number of favorable outcomes is 242.

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

  $P\left(\text{male}\text{and}\text{No}\text{Club}\text{member}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{242}{818}$

Now,

Note that

In the table, 350 of 818 students are non − club members.

In this case, the number of favorable outcomes is 350 and number of possible outcomes is 818.

  $P\left(\text{No}\text{club}\text{member}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{350}{818}$

Apply the conditional probability:

  $\begin{array}{l}P\left(\text{male}|\text{No}\text{club}\text{member}\right)=\frac{P\left(\text{male}\text{and}\text{No}\text{club}\text{member}\right)}{P\left(\text{No}\text{club}\text{member}\right)}\\ =\frac{\frac{242}{818}}{\frac{350}{818}}\\ =\frac{242}{350}\\ =\frac{121}{175}\\ \approx 0.69\end{array}$

Thus,

The conditional probability formale student not a club member is approx. 0.69.