1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
The scary thing is that only 15% of doctors get the answer right. That's not very comforting if you're the one getting a mammography (or whatever screening).
The same problem applies to all these "security" measures that have been put in place, they may catch 80% of the actual terrorists (unlikely) but the policies also yield false positives. The problem is, if you screen 10 million passengers and have 1% false positive rate, that's 100000 false positives you have to sift through - wasting time and money, and not to mention, making 100000 folks very irritated.
Now I'm sure you're all wondering, just what is the answer to the breast cancer question?
7.8%
Here's an explanation: An Intuitive Explanation of Bayesian Reasoning
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