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 Was playing around with neural nets the last couple days, and when I came across this problem it immediately looked very nail-shaped to me. Probably isn't the most efficient tool for the job, but here's my approach:

I trained a neural net to predict success based on ability scores, then ran a few different search algorithms to find the best possible use of 10 points. The ~60m permutations were slightly too many for me to search exhaustively, so I tried predicting on a random sample, using a greedy algorithm to add one point at a time, and another one that added 7 points at a time (maxing out the number of permutations I could fit into memory at a time). 

The best-scoring distribution of stats I could find was CHA: 5, CON: 20, DEX: 13, INT: 13, STR: 6, WIS: 15. According to the calibration curve, that should have a roughly ~75% chance of success.