Probability puzzle

malthrin

I came up with this puzzle after reading Vaniver's excellent post on the Value of Information. I enjoyed working it out over Thanksgiving and thought I'd share it with the rest of you.

Your friend holds up a curiously warped coin. "Let's play a game," he says. "I've tampered with this quarter. It could come up all heads, all tails, or any value in between. I want you to predict a coin flip; if you get it right, I'll pay you $1, and if you're wrong, you pay me $3."

"Absolutely, on one condition," you reply. "We repeat this bet until I decide to stop or we finish N games."

What is the minimum value of N that lets you come out ahead on average?

Each game, you may choose heads or tails, or to end the sequence of bets with that coin. Assume that all heads:tails ratios are equally likely for the coin.

edit: since a couple people have gotten it, I'll link my solution: http://pastebin.com/XsEizNFL

I came up with this puzzle after reading Vaniver's excellent post on the Value of Information. I enjoyed working it out over Thanksgiving and thought I'd share it with the rest of you.

"Absolutely, on one condition," you reply. "We repeat this bet until I decide to stop or we finish N games."

What is the minimum value of N that lets you come out ahead on average?

Each game, you may choose heads or tails, or to end the sequence of bets with that coin. Assume that all heads:tails ratios are equally likely for the coin.

edit: since a couple people have gotten it, I'll link my solution: http://pastebin.com/XsEizNFL

This seems to be a good opportunity to use a trick I figured out for approximating the probability of a binary event

That's just Bayesian updating. P(H|E)=P(H)*P(E|H)/P(E)

P(x)=P(H), x=P(E|H), and the integral of the previous step is P(E), since that's the whole point of your probability function.

So that's why it works... I knew it probably tied into Bayes, but I didn't know exactly how.

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Probability puzzle

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Probability puzzle

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