-- An Example Worth Knowing --

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Medical test diagnoses are just one example (but an important one) of how difficult or tricky it is to understand logical probabilities; in fact it’s well-known that doctors themselves often fail in their understanding of medical test results. One, of many, common examples often given in chapters on Bayesian or conditional statistics runs along these lines:

Suppose 1% of 40-year-old women have breast cancer. And suppose a certain mammography machine correctly diagnoses breast cancer 90% of the time (i.e., IF a woman has breast cancer there is a 90% chance the machine will say so). Suppose the same machine has a 10% chance of giving a false-positive — it says a woman has breast cancer but she does NOT. 

Suppose now, Mary, a 40-year-old woman, goes in for a regular mammography screening and the machine indicates she has breast cancer. What is the probability that she actually does?

People (including doctors) often think there must be close to a 90% chance she is afflicted with cancer. In actuality, it is closer to an 8% chance!

The initial math is not all that difficult:

1% of 40-year-old women have breast cancer, so out of say 1000 such women who go in for testing, ~10 will actually have the disease, on average, and 990 will not.

The machine has 90% accuracy so of those 10 with cancer the machine will diagnose 9 of them correctly (but miss one).
Of the 990 without breast cancer the machine will yield a false-positive on 10% of them, or 99 women.

Thus, out of 1000 original women tested, a total of 108 (9 + 99) will test positive for breast cancer, though only 9 will actually have it.

9 out of 108 (positives) turns out to be a final accuracy rate for a positive result of ~8.3%. THAT is the likelihood, from this one screening, that Mary truly has breast cancer (it is unfortunate how much fear and anxiety such tests automatically generate -- this goes for a number of other medical tests as well). [Note how things would change IF the machine gave NO false-positives (but all do).]

The above is just the basic, rough (but noteworthy) math of the given situation — there are plenty of other variables to consider: any known genetics of the patient or relevant past family history, or current pertinent physical or physiological results. But I’m using the example solely to portray how easily our common sense or intuitions mislead us. The problem is that people automatically assume a "90% accuracy" rate means that any given result has a 90% chance of being true, when in fact a larger context with more conditional factors must be brought into consideration when looking at any single case... and guess what, that's almost always true in life.————————————————

ADDENDUM:   For readers surprised by these numbers I ought further explain that this sort of confusion is commonplace for “screening” tests, which are employed to find candidates for further “diagnostic” testing or examination that is more specific for the condition being investigated (anytime a doctor orders a test for you always worth asking if it is a screening test OR a diagnostic test — perhaps most major ailments these days have both).