What are your chances of having breast cancer? (if you are a woman and have positive results from mammogram)

You are a female who has just received a positive result from a mammogram which you have undergone under the subtle influence of the many advertisements put up by the government. Yes, it is a positive result. You ask the gamemaster (also known as the doctor), what are your chances of actually having breast cancer?

Now how does the gamemaster know? He probably has a manual that details the research conducted on the accuracy of mammograms. It'll say something like this:

   1. The probability that a woman has breast cancer (prevalence) is 1 percent.

   2. If a woman has breast cancer, the probability that she tests positive (sensitivity) is 90 percent.

   3. If a woman does not have breast cancer, the probability that she nonetheless tests positive (false-positive rate) is 9 percent.

And the answer that the gamemaster should infer from the above is one of the following:

   a. The probability that you have breast cancer is about 81 percent.


   b. Out of 10 women with a positive mammogram, about nine have breast cancer.

   c. Out of 10 women with a positive mammogram, about one has breast cancer.

   d. The probability that she has breast cancer is about 1 percent.

What do you think is the probability of having breast cancer given a positive result from a mammogram? To extract brain juice, give it a try before reading on.









The answer is C. Your dice of life has 10 sides, and only one of them says breast cancer. Kind of. The reason is not that you are all guys, but that in interpreting the information given, we tend to make some assumptions.

The answer that most would come up with is B, because we only utilise statement no.2, that "if a woman has breast cancer, the probability that she tests positive (sensitivity) is 90 percent."

In doing so, we are already assuming that you have breast cancer. What if you don't have breast cancer? Then the chances of you still having a positive result is relatively low at 9 percent (false-positive rate), but because a majority of the population does not have breast cancer (see statement 1), that 9 percent becomes a large number of people and you are likely to be one of them. To get the figure of one in ten, we do some simple calculations.

Percentage of population that has breast cancer and tests positive: 90% x 1%(of population) = 0.9%
Percentage of population that does not have breast cancer and tests positive: 9% x 99% = 8.91%
Probability that you test positive and have breast cancer: 0.9% divide by (8.91%+0.91%) = about 10%

Gamemasters should get a degree in statistics too. In case you're wondering, it came from an issue of Scientific American Mind (a magazine), and it called for greater clarity in the presentation of research results, seeing as most gamemasters chose B as well, causing unnecessary worry for patients. Now you know better.