An athlete suspected of having used steroids is given two tests that operate independently of each other. Test A has probability 0.91 of being positive if steroids have been used. Test B has probability 0.77 of being positive if steroids have been used.

and the question is....

.02

To solve this problem, we can use the conditional probability formula. Let's define some variables to make the calculations easier.

Let A represent the event that Test A is positive.
Let B represent the event that Test B is positive.
Let S represent the event that the athlete has used steroids.

We are given the following probabilities:
P(A|S) = 0.91 (Test A is positive given that the athlete has used steroids)
P(B|S) = 0.77 (Test B is positive given that the athlete has used steroids)

We want to find the probability that the athlete has used steroids given that both Test A and Test B are positive, which can be written as P(S|A and B).

To find P(S|A and B), we can use Bayes' theorem:

P(S|A and B) = (P(A and B|S) * P(S)) / P(A and B),

where P(A and B|S) is the probability of both Test A and Test B being positive given that the athlete has used steroids.

We can calculate P(A and B|S) as follows:

P(A and B|S) = P(A|S) * P(B|S) = 0.91 * 0.77 = 0.7017.

Next, we need to calculate P(A and B), which is the probability of both Test A and Test B being positive, regardless of whether the athlete has used steroids.

P(A and B) can be written as:

P(A and B) = P(A and B|S) * P(S) + P(A and B|not S) * P(not S),

where P(not S) is the probability that the athlete has not used steroids.

Since the two tests operate independently, P(A and B|not S) can be calculated as:

P(A and B|not S) = P(A|not S) * P(B|not S),

where P(A|not S) is the probability of Test A being positive given that the athlete has not used steroids, and P(B|not S) is the probability of Test B being positive given that the athlete has not used steroids.

The question does not provide these probabilities, so we need further information or assumptions to continue.

With the additional probabilities, we can substitute them into the equation and calculate P(S|A and B) using Bayes' theorem.