It is known that steroids give users an advantage in athletic contests, but it is also known that steroid use is banned in athletics.
As a result, a steroid testing program has been instituted and athletes are randomly tested. The test procedures are believed to be equally effective between users and non-users. 90% of the athletes affected by this program are clean. The probability that the next athlete will be a user and fail the test is 0.098. Given that an athlete is a user, what is probability the test is accurate?
To find the probability that the test is accurate given that an athlete is a user, we need to use Bayes' theorem. Bayes' theorem allows us to update a probability based on new information.
Let's break down the information provided in the question:
1. The overall prevalence of clean athletes is 90%.
2. The probability that the next athlete will be a user and fail the test is 0.098.
Let's define the events:
A = Athlete is a user
B = Test result is accurate
We are looking for P(B|A), the probability that the test is accurate given that the athlete is a user.
Now, let's calculate:
P(A) = Probability of an athlete being a user = 1 - P(clean athlete) = 1 - 0.9 = 0.1
P(B|A) = Probability that the test is accurate given that the athlete is a user
To calculate P(B|A), we need to know two additional probabilities:
P(A and B) = Probability that an athlete is a user and the test is accurate
P(B) = Probability that the test is accurate
We can calculate P(B) using the Law of Total Probability:
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
Given that the test procedures are believed to be equally effective between users and non-users, the probability that a clean athlete will pass the test (P(B|not A)) is 1, and the probability that an athlete is clean (P(not A)) is 0.9.
P(B) = P(B|A) * 0.1 + 1 * 0.9
P(B) = P(B|A) * 0.1 + 0.9
Now we have:
P(A and B) = P(B|A) * P(A) (this is what we want to find)
To use Bayes' theorem, we substitute the above values into the formula:
P(A and B) = P(B|A) * P(A)
P(B|A) * P(A) = P(B|A) * 0.1
Now we can solve for P(B|A):
P(B|A) = (P(B|A) * 0.1)/(0.098)
Simplifying further:
P(B|A) = 0.1/0.098
So, the probability that the test is accurate given that an athlete is a user is approximately 1.02 or 102%.