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You can always become a matematician in prison and solve one of millennium problems. That's one thing that you can't do if you're dead.

and how often does it happen?

You can certainly do mathematics and publish your work while in prison.

Because the criminal justice system gets things wrong, and we can't undo the death penalty.

That's a choice you might make for yourself, rather than a choice the state makes for you. There's a big difference.