You won the game, but which opening did you play?

Question

Opening A wins 60\% of your games. Opening B wins only 30\%. Before each game you pick one of the two openings at random with a fair coin. You just won. What is the probability you played Opening A?

Accepted Answer

Each opening is equally likely to be chosen. Over 100 games, expect about 50 with each: - Opening A: 50 	imes 0.60 = 30 wins. - Opening B: 50 	imes 0.30 = 15 wins. Of the 30 + 15 = 45 wins, 30 came from Opening A: P(	ext{A} \mid 	ext{win}) = \frac{30}{45} = \frac{2}{3} \approx 0.667. A win is twice as likely to come from Opening A, so your belief climbs from 50 percent to about 67 percent. Takeaway. The winning opening is inferred from the outcome by comparing how well each option explains it, a small Bayesian update.

You won the game, but which opening did you play?

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