Estimates of the prevalence of Alzheimer's disease have...

Estimates of the prevalence of Alzheimer's disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table $3.5 .$ Suppose an unrelated 77 -year-old man, 76 -year-old woman, and 82-year-old woman are selected from a community. Suppose we know one of the three people has Alzheimer's disease, but we don't know which one. What is the conditional probability that the affected person is a woman?

Key Concepts

Bayes’ Theorem

Bayes’ Theorem is a method for updating the probability estimate for a hypothesis based on new evidence. It is particularly useful in this problem for integrating the prior probabilities (from the prevalence data) with the new information that one person has the disease, thereby allowing the computation of the probability that this person is a woman.

Conditional Probability

Conditional probability deals with finding the probability of an event occurring given that another event has already occurred. In this problem, it is used to determine the chance that the Alzheimer’s-affected person is a woman, given the knowledge that exactly one of the three selected individuals has Alzheimer’s disease.

Prevalence

Prevalence is the measure of how common a disease is within a certain population at a specific time. The prevalence estimates provided by Pfeffer et al. are used to gauge the likelihood of Alzheimer’s disease in individuals of different ages, which is fundamental to calculating the specific conditional probabilities in this scenario.

Law of Total Probability

The Law of Total Probability provides a framework for breaking down complicated probability scenarios into simpler, mutually exclusive cases. In this context, it is used to combine the separate probabilities of each individual having Alzheimer’s disease based on their specific attributes (like age and gender) to arrive at an overall probability, which is then utilized in calculating the conditional probability.

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