Chapter 5.3, Problem 62E

(a)

To determine

Probability for male young adult having good chance of much more than middle − class income at age 30.

Answer to Problem 62E

Probability for male young adult young adult having good chance of much more than middle − class income at age 30,

  $P\left(\text{G}|\text{M}\right)\approx 0.3083$

Explanation of Solution

Given information:

Responses are summarized in the two − way table:

  

According to conditional probability,

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

We know

G: Good chance

M: Male

Note that

The information about 4826 young adults is provided in the table.

Thus,

The number of possible outcomes is 4826.

Also note that

In the table, 2459 of the 4826 young adults are male.

Thus,

The number of favorable outcomes is 2459.

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

  $P\left(\text{M}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{2459}{4826}$

Now,

Note that

In the table, 758male adults of the total 4826 young adults have a good chance. In this case, the number of favorable outcomes is 758 and number of possible outcomes is 4826.

  $P\left(\text{G}\text{and}\text{M}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{758}{4826}$

Apply conditional probability:

  $P\left(\text{G}|\text{M}\right)=\frac{P\left(\text{G}\text{and}\text{M}\right)}{P\left(\text{M}\right)}=\frac{\frac{758}{4826}}{\frac{2459}{4826}}=\frac{758}{2459}\approx 0.3083=30.83\%$

Therefore,

Around 30.83% of the young male adults think that there is good chance for them to have much more than middle − class income at age 30 and the probability is 0.3083.

(b)

To determine

Probability for the female respondent didn’t say “almost no chance”.

Answer to Problem 62E

Probability for the female respondent didn’t say “almost no chance”,

  $P\left({\text{M}}^{\text{c}}|{\text{N}}^{\text{c}}\right)\approx 0.4903$

Explanation of Solution

Given information:

Responses are summarized in the two − way table:

  

According to complement rule,

  $P\left({\text{A}}^{\text{c}}\right)=P\left(\text{not}\text{A}\right)=1-P\left(\text{A}\right)$

According to conditional probability,

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

We know

N: Almost no chance

M: Male

Note that

The information about 4826 young adults is provided in the table.

Thus,

The number of possible outcomes is 4826.

Also note that

In the table, 194 of the 4826 young adults have the opinion “Almost no chance”.

That implies

Remaining 4632 young adults do not have the opinion “Almost no chance”.

Thus,

The number of favorable outcomes is 4632.

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

  $P\left({\text{N}}^{\text{c}}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{4632}{4826}$

Now,

Note that

In the table 96 female young adults of the 4826 total young adults have the opinion “Almost no chance”.

Since the total number of female young adults are 2367.

This implies

Remaining 2271 female young adults of the 4826 total young adults did not have the opinion “Almost no chance”

In this case, the number of favorable outcomes is 2271 and the number of possible outcomes is 4826.

  $P\left({\text{M}}^{\text{c}}\text{and}{\text{N}}^{\text{c}}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}}=\frac{2271}{4826}$

Apply conditional probability:

  $P\left({\text{M}}^{\text{c}}|{\text{N}}^{\text{c}}\right)=\frac{P\left({\text{M}}^{\text{c}}\text{and}{\text{N}}^{\text{c}}\right)}{P\left({\text{N}}^{\text{c}}\right)}=\frac{\frac{2271}{4826}}{\frac{4632}{4826}}=\frac{2271}{4632}=\frac{757}{1544}\approx 0.4903=49.03\%$

Therefore,

Around 49.03% young adults didn’t say “almost no chance” are Females and the probability for the female respondent didn’t say “almost no chance” is 0.4903.