We are conducting a test on a rare disease. A positive result means that, according to the test, the subject is infected. The following characteristics are known about the test and the disease: If a person is infected, the person has a $95\%$ chance of testing positive. When a healthy person is tested, the test has a $99\%$ chance of giving a negative result. A mere $0.1\%$ of the population is infected with the disease. If a person is tested positive for the disease. What is the probability that the person actually has the disease? I don't understand how to write the conditional probabilities if someone can explain to me than I should know how to solve it, but I'm not sure how to convert the text into the mathematics language.
closed as off-topic by Xander Henderson, mechanodroid, Did, BruceET, Jam Sep 16 '18 at 23:33
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This involves using the Law of Total Probability and Bayes' Theorem. Let + and - represent positive and negative tests; let D and N represent having the disease and not. Here is an outline to get you started. Try to match each part to formulas in your text.
You seek $P(D|+) = P(D \cap +)/P(+).$
For the numerator, use $P(D \cap +) = P(D)P(+|D),$ where numerical values for both factors are given in the statement of the problem.
For the denominator, start with $P(+) = P(D \cap +) + P(N \cap +).$ You already know the first term from the numerator. Find a similar way to evaluate the second term from information given in the problem.