Question

When a test for steroids is given to soccer players, 98% of the players taking steroids test positive and 12% of the players not taking steroids test positive. Suppose that 5% of soccer players take steroids. What is the probability that a soccer player who tests positive takes steroids?

Short answer

Answer

The probability that a soccer player who tests positive takes steroids is \(0.3006\).

Step-by-step solution

Step-1: Given Information

98% of the players taking steroids test positive and 12% of the players not taking steroids test positive.

Step-2: Definition and Formula  

Bayes’ theorem,

\(P(F|E) = \frac{{P(E|F)P(F)}}{{P(E|F)P(F) + P(E|\overline F )P(\overline F )}}\)

Conditional probability: \(P(B|A) = \frac{{P(A \cap B)}}{{P(A)}}\)

Step-3: Calculated Probability

Let,

E = test Positive

F = use steroids

\(P(F) = 5\% \)

\( = 0.05\)

\(P(E|F) = 98\% \)

\( = 0.98\)

\(P(E|\overline F ) = 12\% \)

\( = 0.12\)

Use the complement rule:

\(P(\overline F ) = 1 - P(F)\)

\( = 1 - 0.05\)

\( = 0.95\)

Using Bayes’ theorem, we get

\(P(F|E) = \frac{{P(E|F)P(F)}}{{P(E|F)P(F) + P(E|\overline F )P(\overline F )}}\)

\( = \frac{{(0.98)(0.05)}}{{(0.98)(0.05) + (0.12)(0.95)}}\)

\( = \frac{{0.049}}{{0.163}}\)

\( = 0.3006\)

Hence,

The probability that a soccer player who tests positive takes steroids is \(0.3006\).