#### Probability: Natural frequencies

Students use natural frequencies to estimate probabilities.

It is much easier for all of us to calculate probabilities when we are speaking of numbers (for example, 8 out of 100) rather than percentages (8%).

In a famous study, Gerd Gigerenzer asked doctors the following question regarding breast cancer screening of women with no symptoms:

“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 not have 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?"

When asked this question, “ninety-five out of 100 [doctors in the U.S.] estimated the probability of breast cancer

to be about 75 percent.” But they were wrong. Answer!

Let’s try the problem again, this time with natural frequencies (with rounding):

“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?” Answer!

Using natural frequencies, the calculation is the same, but the correct answer is much easier to derive.

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