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Posted By: ah_clem
Date: Tuesday, 27 January 2015, at 7:39 p.m.

In Response To: A made-up game to illustrate some points about skill and luck (Timothy Chow)

Here's a simple version of the game to illustrate what might be happening here:

Two players, Red and Blue, flip a coin. If it comes up heads, Red goes to the free-throw line and if he sinks it he wins the game. If it comes up tails, Blue goes to the line and if he sinks it he wins. If the result is a miss for either player, the coin is flipped again.

Now look at what happens in the extreme if Red has 100% accuracy and Blue has 0% accuracy. Blue can never win the game, so Red wins every time. What about the luck?

Half the games are decided on the first coin flip. If it comes up heads, red sinks the basket and wins. So, Red is "luckier" 50% of the time.

If the first toss comes up tails, Blue goes to the line and misses and the coin is tossed again. Half the time it will come up heads and red will win with the flip-count being even. So the luck is even 25% of the time.

If the first two tosses come up tails, then there will be more tails than heads and Blue will be luckier once the eventual heads comes up and the game ends. This happens 25% of the time.

We have the following summary:
Red Luckier : 50%
Equal luck : 25%
Blue Luckier: 25%

What's happening here is that having the sample depend on the game result distorts things - when Red gets lucky the game ends, when blue gets lucky the game continues. Is the same sort of thing going on with Tim's example?

I seem to recall a discussion a few years ago about the distribution - since doubles are more likely to occur on the last roll of the game, breaking up one's samples into games (as opposed to simply looking at all rolls) distorts the dice distribution.

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