Chapter 5.3, Problem 97E

a.

To determine

To find:the percentage of the entire population of being lactose intolerant.

Answer to Problem 97E

The percentage of the entire population of being lactose intolerant is $25.70\%$ .

Explanation of Solution

Given:

  $\begin{array}{l}P\left(\text{White}\right)=82\%=0.82\\ P\left(\text{Black}\right)=14\%=0.14\\ P\left(\text{Asian}\right)=4\%=0.04\end{array}$ .

Concept used:

Multiplication rule $P\left(A\cap B\cap C\right)=P\left(A\right)\times P\left(B\right)\times P\left(C\right)$

Calculation:

According to question:

  $\begin{array}{l}P\left(\text{lactose}|\text{white}\right)=15\%=0.15\\ P\left(\text{lactose}|\text{black}\right)=70\%=0.70\\ P\left(\text{lactose}|\text{Asian}\right)=90\%=0.90\end{array}$

Now, to find the probability of being lactose intolerant for each group:

  $\begin{array}{l}P\left(\text{lactose}\cap \text{white}\right)=P\left(\text{lactose}|\text{white}\right)\times P\left(\text{white}\right)=0.15\times 0.82=0.2464\\ P\left(\text{lactose}\cap \text{black}\right)=P\left(\text{lactose}|\text{black}\right)\times P\left(\text{black}\right)=0.70\times 0.14=0.098\\ P\left(\text{lactose}|\cap \text{Asian}\right)=P\left(\text{lactose}|\text{Asian}\right)\times P\left(\text{Asian}\right)=0.90\times 0.04=0.0360\end{array}$

Add the corresponding probabilities:

  $\begin{array}{l}P\left(\text{lactose}\right)\\ =0.1230+0.0980+0.360\\ =0.2570\\ \Rightarrow P\left(\text{lactose}\right)=0.2570\times 100=25.70\%\end{array}$

Hence, the percentage of the entire population of being lactose intolerant is $25.70\%$ .

b.

To determine

To find: the percentage of people whose are lactose intolerant are Asian.

Answer to Problem 97E

  $P\left(\text{Asian}|\text{lactose}\right)=14.01\%$

Explanation of Solution

Given:

  $\begin{array}{l}P\left(\text{White}\right)=82\%=0.82\\ P\left(\text{Black}\right)=14\%=0.14\\ P\left(\text{Asian}\right)=4\%=0.04\end{array}$ .

Concept used:

Definition conditional probability:

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

Calculation:

From part (a): $P\left(\text{lactose}\right)=0.2570$

Definition conditional probability:

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

Thus obtained:

  $\begin{array}{l}P\left(\text{Asian}|\text{lactose}\right)=\frac{P\left(\text{lactose}\cap \text{Asian}\right)}{P\left(\text{Asian}\right)}\\ =\frac{0.360}{0.2570}\approx 0.1401=14.01\%\end{array}$

Hence,the percentage of people whose are lactose intolerant are Asian is $14.01\%$ .