Question

Suppose there is a 26.6% probability that a randomly selected person aged 35 years or older is a jogger. In addition, there is a 22.6% probability that a randomly selected person aged 35 years or older is female, given that he or she jogs. What is the probability that a randomly selected person aged 35 years or older is female and jogs? Would it be unusual to randomly select a person aged 35 years or older who is female and jogs? The probability that a randomly selected person aged 35 years or older is female and jogs is Would it be unusual? No Yes

Answer 1

There is a 6.01% probability that a randomly selected person aged 40 years or older is a female and jogs.

It would not be unusual to randomly select a person aged 40 years or older who is a female and jogs.

Explanation:

We have these following probabilities:

A 26.6% probability that a randomly selected person aged 35 years or older is a jogger, so
`P(A) = 0.266`.

A 22.6% probability that a randomly selected person aged 35 years or older is female, given that he or she jogs. I am going to say that P(B) is the probability that the person is a female. P(B/A) is the probability that the person is a female, given that he/she jogs. So
`P(B/A) = 0.226`.

The Bayes theorem states that:

`P(B/A) = (P(A \cap B))/(P(A))`

In which
`P(A \cap B)` is the probability that the person does both thigs, so, in this problem, the probability that a randomly selected person aged 40 years or older is a female and jogs.

So

`P(A \cap B) = P(A).P(B/A) = 0.266*0.226 = 0.0601`

There is a 6.01% probability that a randomly selected person aged 40 years or older is a female and jogs.

A probability is unusual when it is smaller than 5%.

So it would not be unusual to randomly select a person aged 40 years or older who is a female and jogs.