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. Assuming you have zero knowledge of your opponent's skill level and that, as with all other parts of this problem, you're equally likely to play an amateur, an equal, or a professional, what's the expected number of games you'll win? You should, of course, assume you play the same opponent all three times. Express your general answer in terms of p1 and p2, but verify your answer using p1=0.7 and p2=0.25 out to 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.
Assuming you have zero knowledge of your opponent's skill level and that, as with all other parts of this problem, you're equally likely to play an amateur, an equal, or a professional, what's the expected number of games you'll win? You should, of course, assume you play the same opponent all three times. Express your general answer in terms of p1 and p2, but verify your answer using p1=0.7 and p2=0.25 out to three decimal places.
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