You've been scheduled to play three games of pickleball against someone you know very little about. Assume your opponent is either an amateur, an equal (i.e., they are just as talented as you are, but no more), or a professional. You can assume your opponent is equally likely to be any of the three levels, and that you'll win a single game against the amateur, your equal, and the professional with probabilities p1, 0.5, and p2 respectively, where we'll assume that 0<p2<0.5<p1<1. Given that you win all three games, what's the probability that your opponent was the professional? Express your general answer in terms of p1 and p2, and verify your answer using p1=0.7 and p2=0.25 out to at least three decimal places.
You've been scheduled to play three games of pickleball against someone you know very little about. Assume your opponent is either an amateur, an equal (i.e., they are just as talented as you are, but no more), or a professional. You can assume your opponent is equally likely to be any of the three levels, and that you'll win a single game against the amateur, your equal, and the professional with probabilities p1, 0.5, and p2 respectively, where we'll assume that 0<p2<0.5<p1<1.
Given that you win all three games, what's the
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