Question

Lactose intolerance causes difficulty digesting dairy products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States (ignoring other groups and people who consider themselves to belong to more than one race), 82% of the population is white, 14% is black, and 4% is Asian. Moreover, 15% of whites, 70% of blacks, and 90% of Asians are lactose intolerant. Step 1:

What percent of the entire population is lactose intolerant?
Give your answer in percents to 1 decimal place (e.g., 22.4).

Step 2:
What percent of people who are lactose intolerant are Asian?
Give your answer in percents as a whole number (e.g., 20).

Answer 1

The percentage of the entire population in lactose intolerant is 25.7% The percentage of people who are lactose intolerant is 14%

Based on the data that follows

The subject of Event W is White Americans.

The subject of Event B is Black Americans.

The topic of Event A is Asian Americans in the US.

The topic of Event L is lactose intolerance.

Afterwards, the probabilities that are

P(W) = 0.82

P(B) = 0.14

P(A) = 0.04

Now Conditional probability of Lactose intolerant for White, Black, and Asian people is

P(L/W) = 0.15

P(LIB) = 0.70

P(LA) = 0.90

1)

By summing up the probabilities of lactose intolerance in each racial group and weighting them according to their respective population proportions, we can use the conditional probability approach to determine the chance of lactose intolerance in the overall population.

The probability of the entire population in lactose intolerant is given by

P(L) = P(W)P(LW) + P(B)P(LB) + P(A)P(LA)

= 0.82 × 0.15 +0.14 x 0.70 +0.04 x 0.90 0.257

The percentage of the entire population in lactose intolerant is 25.7%

2)

By multiplying the conditional likelihood of being Asian given lactose intolerance by the prior probability of being Asian and dividing the result by the prior probability of lactose intolerance, we can use the Bayes theorem to get the probability of being Asian given lactose intolerance.

The probability of people who are lactose intolerant are Asian is

P(A|L) =
`(P(A)P(L|A))/(P(A))`

P(A|L) =
`(0.04 (0.90))/(0.257)`

P(A|L) = 0.140077

The percentage of people who are lactose intolerant are Asian is 14%

Answer 2

1) 25.7% of the population is lactose intolerant.

2) 14% of the lactose intolerant people is Asian

Explanation:

Population distribution according to race:

• 82% white
• 14% black
• 4% Asian

Lactose intolerance of races

• 15% of whites
• 70% of blacks
• 90% of Asians

Assume US population consists of 1000 people.

820 people are white, 140 people are black and 40 people are Asian.

Then 15% of 820 white, 70% of black, 90% of Asian is lactose intolerant.

• 15% of 820 is 123, which gives the white lactose intolerant people

• 70% of 140 is 98, which gives the black lactose intolerant people

• 90% of 40 is 36, which gives the Asian lactose intolerant people.

Step1

Then total lactose intolerant people in the US population is 123+98+36=257

Total population is 1000 people, therefore
`(257)/(1000)`=25.7% of the population is lactose intolerant.

Step2

There are 36 Asian lactose intolerant people. Total lactose intolerant people is 257.
`(36)/(257)` = 14% of them is Asian.