I think you're misreading that study. It compares the number of women estimated to have their life prolonged because of a mammogram, to the number of women estimated to have been treated unnecessarily because of a mammogram. The issue with whether mammograms prolong life is different than the issue of whether mammograms accurately predict cancer. Depending what research you look at, other approaches are sometimes seen as just as effective as mammograms. This is very controversial right now among people who do cancer treatment.

The point is, however, the article you cite does not say for every correct treatment there are 10 unneeded treatments; it suggests that for every life saved there may be 10 unneeded treatments. The summary doesn't say what they mean by a life saved. Mammograms find a certain number of cancers. Other diagnostic techniques also find a certain number of cancers. These sets overlap. It may be that the authors consider the number of lives saved by mammograms to be the delta between cancers detected/treated due to mammograms and due to other diagnostic techniques. This seems to me to be a valid issue, but it has nothing to do with how accurate mammograms are in detecting actual cancers, which seems to be 80%. (So, for example, what if other diagnostic techniques have a 70% chance of detecting cancer -- this might lead to the 10 unneeded treatments per one life saved statistic)

The issue I have with the article posted above is that it claims the probability of a woman having cancer after a positive mammogram is 7.8% when actual results show this to be 80%.

I did contact a statistics site that had a similar statement about AIDS tests and their response was, Baysian statistics are correct and they saw no problem with telling people that a positive AIDS test only indicated a very low chance of having AIDS.