Question

Get rich A survey of $4826$ randomly selected young adults (aged $19$ 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 shows the responses.

$14$ Choose a survey respondent at random.

(a) Given that the person selected is male, what’s the probability that he answered “almost certain”?

(b) If the person selected said “some chance but probably not,” what’s the probability that the person is female?

Short answer

Part (a) P (Almost certain/Male) =$0.2428$

Part (b) P (Female/some chance)=$0.5983$

Step-by-step solution

Part (a) Step 1. Given Information 

$N=4826$ young adults were chosen as part of the sample. The responses are organised in a two-way table.

Part (a) Step 2. Concept Used

Formula's used:$P(A)=\frac{totalnumberofoutcomescorrespondingtoeventA}{numberofoutcomesinsamplespace}$

Part (a) Step 3. Calculation

The selected person is a man, according to the inquiry. Now we must calculate the chance that the result for "nearly certain" is correct.

Therefore,$P(AlmostcertainandMale)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{597}{4826}$

$P\left(Male\right)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{2459}{4826}$

As a result, the conditional probability is:

$$\begin{gathered} P(Almostcertain|Male)=\frac{P(AlmostcertainandMale)}{P\left(Male\right)} \\ P(Almostcertain|Male)=\frac{{\displaystyle \frac{597}{4826}}}{{\displaystyle \frac{2459}{4856}}} \\ P(Almostcertain|Male)=\frac{597}{2459} \\ P(Almostcertain|Male)\approx 0.2428 \end{gathered}$$

As a result, the likelihood of the result being "nearly definite" is

$P(Almostcertain/Male)=0.2428$

Part (b) Step 1. Calculation

The chosen person is female, according to the query. The next step is to calculate the probability that the result for "some chance but probability not" is correct. Therefore,

$P(Femaleandsomechance)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{426}{4826}$

$P(somechance)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{712}{4826}$

As a result, the conditional probability is:

$$\begin{gathered} P(Female/somechance)=\frac{P(Femaleandsomechance)}{P\left(Female\right)} \\ P(Female/somechance)=\frac{{\displaystyle \frac{426}{4826}}}{{\displaystyle \frac{712}{4826}}} \\ P(Female/somechance)=\frac{426}{712} \\ P(Female/somechance)\approx 0.5983 \end{gathered}$$

As a result, the probability that the result for "some chance but probability not" is "some chance but probability not" is $P(Female/somechance)=0.5983$