Home > Mathematics, Probability, Steven Strogatz, The New York Times > 85% Will Be Smarter After Reading This… If My Calculations Are Right.

85% Will Be Smarter After Reading This… If My Calculations Are Right.

I spotlighted Steven Strogatz’s New York Times column before and those that took my advice to follow him are being rewarded lesson after lesson. In his most recent installment, he introduces one of my favorite mathematical topics: probability. The counterintuitiveness adds to its intrigue and to the necessity of focusing on it in our schools. The other week a few friends and I spent most of our work day trying unsuccessfully to figure out what percentage chance the Bruins had of landing the 1st or 2nd overall pick in the draft. Eventually I just had my mother contact her colleague: an advanced statistics teacher. Thanks for the help! He and Strogatz, as with all good educators, make the opaque clearer. Ok readers, here is the first problem:

The probability that one of these women has breast cancer is 0.8 percent.  If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram.  If a woman does nothave breast cancer, the probability is 7 percent that she will still have a positive mammogram.  Imagine a woman who has a positive mammogram.  What is the probability that she actually has breast cancer?

Let me guess. You’re probably like me and have no idea. Well you’re with at least 95 out of 100 of American doctors. Strogatz now reframes the problem in terms of “natural frequencies” instead of percentages. (Have your answers before continuing)



Eight out of every 1,000 women have breast cancer.  Of these 8 women with breast cancer, 7 will have a positive mammogram.  Of the remaining 992 women who don’t have breast cancer, some 70 will still have a positive mammogram.  Imagine a sample of women who have positive mammograms in screening.  How many of these women actually have breast cancer?

Since a total of 7 + 70 = 77 women have positive mammograms, and only 7 of them truly have breast cancer, the probability of having breast cancer given a positive mammogram is 7 out of 77, which is 1 in 11, or about 9 percent.

It is well worth reading his whole post – the importance of this topic seems to be in inverse proportion to the coverage it receives. I hope to correct that. For my law loving readers, he shares some fresh probabilistic thinking on the OJ trial. Also, just enjoy this line from Strogatz – it makes me think of this blog in many ways. 

So we sacrificed a little precision for a lot of clarity.

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