- Your quiz today is on probability.
- Here is the link for students to use to download their own copy of Mathematica
- Today we consider the breast cancer problem:
"Gigerenzer and his colleagues asked doctors in Germany and the United States to estimate the probability that a woman with a positive mammogram actually has breast cancer, even though she's in a low-risk group: 40 to 50 years old, with no symptoms or family history of breast cancer.
"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?"
We'll use "natural frequencies" to solve this problem.
- Let's start by considering the universe. The best way to represent it is as a table.
- Then we write down our givens:
-
- Then we solve for the unknown, which is
We can also fill in the rest of our table, and do other probabilities.
- Notice that our author estimates -- rounds -- answers, to make his
life a little easier. We'll try that first, then calculate the exact
probabilities. Often the approximations are sufficiently good.
- Now that we've calculated the exact values, what do you think of
this quote from our text:
"As for the American doctors, ninety-five out of a hundred estimated the woman's probability of having breast cancer to be somewhere around 75 percent."
- First of all, nitpicky me, you see why I'm grumpy....
- But, more importantly, what are the consequences for health care?
- Links:
Website maintained by Andy Long. Comments appreciated.