Saturday, February 04, 2006

The Perils of Not Knowing Basic Probability Theory

On last Thursday’s Survivor, three team members had to decide which of them would be stranded on Exile Island. They wanted to do it randomly, so they agreed to play rock-paper-scissors. First, Sally played Misty and Misty lost. Then Misty played Courtney and Misty lost again. So Misty got exiled.

Courtney got lucky. But if she’d had the slightest knowledge of probability, she’d have realized the procedure disadvantaged her. Misty had to lose twice in order to be exiled; Sally also would have had to lose twice. Courtney, however, would only have had to lose once. The procedure they used gave Courtney a 50% chance of being exiled, and the other two only 25% each.

How might they have generated a true 1/3 probability for each contestant to be exiled? These two old posts of mine discuss several possible (isomorphic) answers – just substitute rock-paper-scissors for the coin flip. Or they could have just drawn straws.

Now here’s the question of strategy. Say you were one of those three contestants. And say you understood the probability issues discussed above. What would you do? My first impulse would have been to dispute the mechanism and propose a fair one. But I hope I would have resisted that impulse, and instead simply made sure I took part in the first round of rock-paper-scissors.


Jeff Brown said...

I agree. In a functioning economy, there might be reasons to seek equitable outcomes at one's own expense when the beneficiaries would not have immediately recognized any harm to them -- but I can't think of advantages to such behavior on Survivor.

[Unrelated: I think I had to verify this same string of text earlier.]

Ben said...

This is not necessarily true, if Courtney had lost on the second round, the game could have continued with Courtney playing Sally. The rules would have been:
The first player to get two defeats is exiled.

After the first round, the loser of the first round played again so to maximise the chances of finding the loser immediately. This is a version of Glen's solution from “The Three-Sided Coin: Answers”.

I just realized that it is faster than you claimed because we don’t need the full three rounds in a game. I’m not too sure how to compute the probabilities though.

Glen Whitman said...

Ben -- Good point, their actions were consistent with one possible fair mechanism. But I highly doubt that was the plan, because they didn't discuss it at all. The mechanism you describe is unusual enough that it would presumably require some discussion beforehand.

And you're right that my mechanism for the 3-sided coin can be sped up by foregoing the third flip in a round where one option has already won twice. But that won't happen every time, so it still won't be as fast as the alternative mechanism that has only two flips each round.