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BGonline.org Forums
Simple Probability Problem
Posted By: Timothy Chow In Response To: Simple Probability Problem (Rick Janowski)
Date: Thursday, 27 December 2012, at 6:55 p.m.
To me it seems that percentile and probability should be the same to first order. As Tom put it, if we're playing a game like, "Let's see who's taller!" then the outcome of every contest is (essentially) deterministic. The taller player always wins. If you're in the 75th percentile, then you will win 75% of your matches against the field, assuming you play everyone equally often.
Now imagine gradually introducing a little bit of luck. To first order, everything is the same as before except for the people who are close to you in the percentile ranking: you'll always lose to the people who are in the 77th percentile or higher and you'll always win against those who are in the 73rd percentile or lower. Things might be fuzzy in the interval between 73 and 77 but you'll still win close to 75% of all your matches against the field.
If you add a lot of luck then the fuzz starts to become non-negligible. Still, the first-order approximation is that you'll beat the people weaker than you and you'll lose to the people stronger than you. However, we do have to consider second-order effects. There are more people worse than you than there are people better than you, so typically you will lose more unlucky matches to weaker players than you will win lucky matches against stronger players. Thus your win probability against the field will drop below 75%. Exactly how much, though, depends on what modeling assumptions you introduce about the outcomes of contests.
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