Conditional probability: an easier way

Conditional probabilities are bane of many students of Statistics, but statements of conditional probability come up surprisingly often in real life. For example, as Steven Strogatz writes in the New York Times, when doctors are asked to estimate the probability that a woman has breast cancer given a positive mammogram test result, most get the answer wildly wrong despite being given the population frequency of breast cancer and the conditional probability of false positives from a mammogram test. Here’s one doctor’s experience trying to come up with a number:

“[He] was visibly nervous while trying to figure out what he would tell the woman.  After mulling the numbers over, he finally estimated the woman’s probability of having breast cancer, given that she has a positive mammogram, to be 90 percent.  Nervously, he added, ‘Oh, what nonsense.  I can’t do this.  You should test my daughter; she is studying medicine.’  He knew that his estimate was wrong, but he did not know how to reason better.  Despite the fact that he had spent 10 minutes wringing his mind for an answer, he could not figure out how to draw a sound inference from the probabilities. …

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