Solutions to Bayes and the False Positives Problem – in a nutshell!

ab/ [ab+c(1-a)]

a is the prior probability, i.e. the probability that a hypothesis is true before you see the new evidence. Before the new evidence (the test), this chance is estimated at 1 in 100 (0.01), as we are told that 1 per cent of the people who visit his surgery have the virus. So, a = 0.01

b is the probability of the new evidence if the hypothesis is true. The probability of the new evidence (the positive result on the test) if the hypothesis is true (the patient is sick) is 95%, since the test is 95% accurate. So, b =0.95

c is the probability of the new evidence if the hypothesis is false. The probability of the new evidence (the positive result on the test) if the hypothesis is false (the patient is not sick) is 5% (because the test is 95% accurate, and we can only expect a false positive 5 times in 100). So, c = 0.05

Using Bayes’ Theorem, the updated (posterior) probability = 0.01 x 0.95/ [(0.01 x 0.95) + 0.05 (1-0.01)] = 0.0095 / (0.0095 + 0.0495) = 0.161

So the chance the doctor should give to the patient having the flu, if testing positive, is 16.1 per cent.

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