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Screaming Toes: More than an ice breaker, it's probability in practice!

How to play:
Get your group together in a big circle, then everyone looks at someone else's toes, and then everyone looks up, if the person you're looking at is looking at you, you both have to scream. Naturally, two person screaming toes has the highest probability of a scream, whereas one person screaming toes has the lowest probability. One person screaming toes with a mirror might be interesting, though I will not speculate further.

Now to the more interesting issue, given a set of N people who randomly look at other people's toes (no person's toes are more attractive than another's), on average, how many screams should occur per turn? Here is my solution, though it's probably wrong:

The event space, E = nCr(N,2)
The sample space, S = N!

Thus, the probability is nCr(N,2)/N! that a given pair will scream and, on average, nCr(N,2)/N! * N people will scream per game.

As it turns out, both 2 and 3 person games have the same probability, .5, (which, remember, is the probability two people will scream) and thus more screams will happen in the 3 person game.
Don't ask me how this is a game.

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