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If 2,000 women are screened regularly for ten years, one will benefit from the screening, as she will avoid dying from breast cancer. At the same time, ten healthy women will...become cancer patients and will be treated unnecessarily. These women will have either a part of their breast or the whole breast removed, and they will often receive radiotherapy, and sometimes chemotherapy. Furthermore, about 200 healthy women will experience a false alarm. The psychological strain until one knows whether or not it was cancer, and even afterward, can be severe.
and
It has not been shown that women who undergo regular screening live longer than those who don’t.
Note that we're talking here about scheduled mammograms, done because the calendar shows a particular date, not a diagnostic test done because there are observed symptoms that need to be understood. We're also talking about the general population, not people whose family histories or other risk factors would lead them to be in a higher-risk pool.
For men, you can make a similar argument about PSA tests and prostate cancer though I don't have the numbers handy.
I realize that trying to get people to understand false positives/false negatives and the like is tilting at peculiar windmills, but I do it nonetheless.