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It’s estimated that 50% of women who have reached adulthood since 1973 have had at least one abortion. Wouldn’t that mean that 50% of women had serious pre-existing psychological problems?

Estimated by whom? I believe that number is used by abortion advocates who want to inflate the numbers so as to promote the illusion that abortion is a common method of birth control used by many women. They use that illusion to try to convince more women to have abortions.

However, if you look at the numbers, the 50% estimate just doesn't hold up. Rough and dirty estimate, ignoring confounders such as number of women in childbearing age, statistical analyses, and so forth:

Total number of people in the US: ~300 million.
Total number of women: about half that, or ~150 million.
Total number of abortions committed since 1973: ~50 million.
Percentage of abortions performed on women who have already had at least one abortion: ~50%.

So, the number of women who have had abortions is roughly (50 million / 150 million) x 0.5 = 17%.