Question

Get rich A survey of 4826 randomly selected young adults (aged $19\mathrm{to}25$) asked, “What do you think are the chances you will have much more than a middle-class income at age $30$?” The two-way table summarizes the responses.

Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance.

a. Find $P(G|M)$. Interpret this value in context.

b. Given that the chosen survey respondent didn’t say “almost no chance,” what’s the probability that this person is female? Write your answer as a probability statement using correct symbols for the events.

Short answer

a. Required probability is $P(\mathrm{G}\mid \mathrm{M})\approx 0.3083$

b. Probability of female respondent didn't say "almost no chance" is$P\left({\mathrm{M}}^{\mathrm{c}}\mid {\mathrm{N}}^{\mathrm{c}}\right)\approx 0.4903$

Step-by-step solution

Given Information

It is given that:

Determining probability for male young adult having good chance of much more than middle class income at age 30

Using Conditional Probability: $P(\mathrm{B}\mid \mathrm{A})=\frac{P(\mathrm{A}\cap \mathrm{B})}{P(\mathrm{A})}=\frac{P(\mathrm{A}\text{and}\mathrm{B})}{P(\mathrm{A})}$

G: Good Chance

M: Male

Information about $4826$young adults is provided.

From table, $2459/4826$young adults are males.

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

Also, $78/4826$young adults have a good chance.

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

Using Conditional Probability

$P(\mathrm{G}\mid \mathrm{M})=\frac{P\left(\mathrm{GandM}\right)}{P\left(\mathrm{M}\right)}=\frac{\frac{758}{4826}}{\frac{2259}{4826}}=\frac{758}{2459}\approx 0.3083=30.83\%$

Hence, $30.83\%$of young males are of the view that there is good chance for them to have much more than middle aged income at$30$and probability is$0.3083$

Determining probability of the female respondent didn't say "almost no chance".

As per complement rule,

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

and conditional probability $P(\mathrm{B}\mid \mathrm{A})=\frac{P(\mathrm{A}\cap \mathrm{B})}{P(\mathrm{A})}=\frac{P(\mathrm{A}\text{and}\mathrm{B})}{P(\mathrm{A})}$

N: Almost no chance

M: Male

From table, $194/4826$young adults think that "Almost no chance".

$\Rightarrow 4632$young adults do not have such opinion.

$P\left({\mathrm{N}}^{\mathrm{c}}\right)=\frac{\text{Numberof favourable outcomes}}{\text{Numberof possibleoutcomes}}=\frac{4632}{4826}$

From table, $96/4826$female adults have opinion "Almost no Chance".

As total female young adults are $2367$.

$\Rightarrow 2271/4826$female young adults dis not have above opinion.

$\Rightarrow P\left({\mathrm{M}}^{\mathrm{c}}\text{and}{\mathrm{N}}^{\mathrm{c}}\right)=\frac{\text{Numberof favourableoutcomes}}{\text{Number of possible}}=\frac{2271}{4826}$

Using Conditional Probability:

$P\left({\mathrm{M}}^{\mathrm{c}}\mid {\mathrm{N}}^{\mathrm{c}}\right)=\frac{P\left({\mathrm{M}}^{\mathrm{c}}\text{and}{\mathrm{N}}^{\mathrm{c}}\right)}{P\left({\mathrm{N}}^{\mathrm{c}}\right)}=\frac{\frac{2271}{4226}}{\frac{4632}{4826}}=\frac{2271}{4632}=\frac{757}{1544}\approx 0.4903=49.03\%$

Hence, probability for the female respondent didn't say "almost no chance" is$0.4903$