Question

Here are some statistics collected by a doctor about patients who walk into her office.

∙ 25% of the patients have the flu.

∙ Among patients with the flu, 75% have a fever.

∙ Among patients who don't have the flu, 50% have a fever.

A new person walks into the doctor's office and turns out to have a fever. What is the probability that he has the flu?

Answer 1

Answer 2

To find the probability that the new person who has a fever actually has the flu, we can use Bayes' theorem.

Let's define the events: A: Having the flu B: Having a fever

We are given the following information: P(A) = 0.25 (probability of having the flu) P(B|A) = 0.75 (probability of having a fever given that the person has the flu) P(B|not A) = 0.50 (probability of having a fever given that the person does not have the flu)

We need to find P(A|B), which is the probability of having the flu given that the person has a fever.

Using Bayes' theorem, we have: P(A|B) = (P(B|A) * P(A)) / P(B)

To find P(B), we can use the law of total probability: P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)

P(not A) represents the complement of having the flu, which is 1 - P(A).

Substituting the given values, we can calculate the probabilities:

P(B) = P(B|A) * P(A) + P(B|not A) * P(not A) = 0.75 * 0.25 + 0.50 * (1 - 0.25) = 0.1875 + 0.375 = 0.5625

Now we can calculate P(A|B) using Bayes' theorem:

P(A|B) = (P(B|A) * P(A)) / P(B) = (0.75 * 0.25) / 0.5625 = 0.1875 / 0.5625 = 0.3333

Therefore, the probability that the new person who has a fever actually has the flu is approximately 0.3333 or 33.33%.