Chapter 5.3, Problem 84E

To determine

Probability for the randomly selected U.S person is lactose intolerant Asian.

Answer to Problem 84E

Probability that the randomly selected U.S person is lactose intolerant Asian is approx. 0.1401.

Explanation of Solution

Given information:

Data for African and Asian ancestry in United States:

82% of the population is White

14% of the population is Black

4% of the population is Asian

Lactose intolerant people:

15% are Whites

70% are Blacks

90% are Asians

Calculations:

Probability for White,

  $P\left(\text{W}\right)=0.82$

Probability for Black,

  $P\left(\text{B}\right)=0.14$

Probability for Asian,

  $P\left(\text{A}\right)=0.04$

Probability for White lactose intolerant,

  $P\left(\text{L}|\text{W}\right)=0.15$

Probability for Black lactose intolerant,

  $P\left(\text{L}|\text{B}\right)=0.70$

Probability for Asian lactose intolerant,

  $P\left(\text{L}|\text{A}\right)=0.90$

Apply general multiplication rule:

Probability for lactose intolerant and White,

  $P\left(\text{L}\text{and}\text{W}\right)=P\left(\text{W}\right)\times P\left(\text{L}|\text{W}\right)=0.82\times 0.15=0.123$

Probability for lactose intolerant and Black,

  $P\left(\text{L}\text{and}\text{B}\right)=P\left(\text{B}\right)\times P\left(\text{L}|\text{B}\right)=0.14\times 0.70=0.098$

Probability for lactose intolerant and Asian,

  $P\left(\text{L}\text{and}\text{A}\right)=P\left(\text{A}\right)\times P\left(\text{L}|\text{A}\right)=0.04\times 0.90=0.036$

Since the group of people considering themselves to belong to more than one race are not included.

Apply general addition rule for mutually exclusive events:

  $\begin{array}{l}P\left(\text{L}\right)=P\left(\text{L}\text{and}\text{W}\right)+P\left(\text{L}\text{and}\text{B}\right)+P\left(\text{L}\text{and}\text{A}\right)\\ =0.123+0.098+0.036\\ =0.257\end{array}$

Using conditional probability definition:

  $P\left(\text{A}|\text{L}\right)=\frac{P\left(\text{L}\text{and}\text{A}\right)}{P\left(\text{L}\right)}=\frac{0.036}{0.257}=\frac{36}{257}\approx 0.1401$

Thus,

The conditional probability for randomly selected person is lactose intolerant Asian is approx. 0.1401.