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?
Here's an explanation: An Intuitive Explanation of Bayesian Reasoning