小组

Consider mammographic screening for breast cancer. Using made up percentages for the sake of simplifying the numbers, let’s assume that five in 100 women have breast cancer. Suppose that if a person has breast cancer, then the mammograph test will find it 80 times out of 100. When the test comes out suggesting that breast cancer is present, we say that the result is positive, although of course there is nothing positive about this for the person being tested. (A technical way of saying this is that the sensitivity of the test is 80%.)

The test may also fail in the other direction, namely to indicate breast cancer when none exists. This is called a false positive finding. Suppose that if the person being tested actually doesn’t have breast cancer, the chances that the test nevertheless comes out positive are 10 in 100.

Consider the above breast cancer scenario. An average woman takes the mammograph test and gets a positive test result suggesting breast cancer. What do you think are the odds that she has breast cancer given the observation that the test is positive?

First, use your intuition without applying the Bayes rule, and write down on a piece of paper (not in the answer box below) what you think the chances of having breast cancer are after a positive test result. The intuitive answer will not be a part of your answer. It will be just for your own information.

Next, calculate the posterior odds for her having breast cancer using the Bayes rule. This will be your answer.

Hints:

1. 1、Start by calculating the prior odds.
2. 2、Determine the probability of the observation in case of the event (cancer).
3. 3、Determine the probability of the observation in case of no event (no cancer).
4. 4、Obtain the likelihood ratio as the ratio of the above two probabilities.
5. 5、Finally, multiply the prior odds by the likelihood ratio.

我错在了最不应该错的第一点上。 就怪我想太多。。。。。