Lactose intolerance Lactose intolerance causes difficulty in...

Lactose intolerance Lactose intolerance causes difficulty in 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. ${ }^{19}$ Suppose we select a U.S. person at random. (a) What is the probability that the person is lactose intolerant? Show your work. (b) Given that the person is lactose intolerant, find the probability that he or she is Asian. Show your work.

Lactose intolerance Lactose intolerance causes difficulty in 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. ${ }^{19}$ Suppose we select a U.S. person at random.
(a) What is the probability that the person is lactose intolerant? Show your work.
(b) Given that the person is lactose intolerant, find the probability that he or she is Asian. Show your work.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Law of Total Probability

This concept involves breaking down overall probability into parts that depend on mutually exclusive cases. In the context of the problem, the overall probability that a randomly selected person is lactose intolerant is computed by considering the probabilities for different racial groups, weighted according to their proportion in the population.

Conditional Probability

This refers to the probability of an event occurring given that another event has occurred. In this problem, it is used to find the likelihood that a person is of a certain race, given that they are lactose intolerant.

Bayes' Theorem

Bayes' Theorem provides a way to update prior probability estimates based on new information. Here, it is applied to reverse the conditional probabilities—calculating the probability that an individual belongs to a specific racial group given that they are lactose intolerant—by combining the initial likelihoods of each group and their respective lactose intolerance rates.

Transcript

Please give Ace some feedback