Question

Today, you want to go on a run. The probability of the day being windy is 52%. The probability of going for a run when it is windy is 23%. What is the probability of going on a run, given that it is a windy day? (1 point) the answer is not 44.23 or 23

Answer 1

To solve this problem, you can use Bayes' theorem. Bayes' theorem states:

P(A|B) = P(B|A) * P(A) / P(B)

where:
- P(A|B) is the probability of event A given event B
- P(B|A) is the probability of event B given event A
- P(A) is the probability of event A
- P(B) is the probability of event B

In this case, the events are:
- A: Going on a run
- B: It is a windy day

Given:
- P(B) = 0.52 (probability of a windy day)
- P(A|B) = 0.23 (probability of going for a run on a windy day)
- P(B|A) = P(A and B) / P(A) = P(A and B) / P(A and B) + P(A and not B) = 0.23 / 0.52

Now plug these values into Bayes' theorem:

P(A|B) = P(B|A) * P(A) / P(B)
P(A|B) = (0.23 / 0.52) * P(A) / 0.52
P(A|B) = 0.23 * P(A)

Since the sum of the probabilities must equal 1, you can also express the sum of the probabilities of running on windy and non-windy days as:

P(A) = P(A|B) * P(B) + P(A|not B) * P(not B)
P(A) = 0.23 * 0.52 + P(A|not B) * (1 - 0.52)
P(A) = 0.1196 + P(A|not B) * 0.48

Substitute this back into the Bayes' theorem equation to solve for P(A|B). So the answer should be in the form of 0.23 * (0.52 / 0.1196 + P(A|not B) * 0.48).