The Bent Coin

[jason1990]

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It's like saying if you have AA vs. KK preflop that you don't have enough information to know if you're a favorite or not (maybe there's a 100% chance that a K will flop).

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The reason you have enough information in this case is because the deck was shuffled. The shuffle may not be perfect, but as a practical matter it is usually reasonable to assume that it is, which justifies all of the ordinary probability calculations we do in cases like this.

But suppose you walk into a bar and see the AA and KK on the table, with the rest of the deck in the center. Someone offers to take the KK side and bet against you. I hope you would not accept this bet without first demanding to inspect the rest of the deck and then have it thoroughly shuffled.

When we assume that the next card is equally likely to be any of the unseen cards, we must have a reason to assume this. Complete ignorance is not a reason.

[chezlaw]

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If you are trying to distinguish between events that have not yet happened and those that have, here is how I would undistinguish it. Assume the future event already happened and the two of you are gambling on a videotape. And what if neither one of you KNOW whether the KK vs AA hand is live or not?

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That's not the distinction I'm making. The zillionth digit in the (decimal) expansion is fixed in time but it's an unneccesary confusion to say there is a 10% chance of it being a three rather than if we guess unknown (to us) expansion digits we will get it right 10% of the time.

Unless you claim its not fixed until someone knows what it is. Some sort of probability field collapsing.

chez

[surftheiop]

DS- could this be a comparable analogy?
Arent you basically saying that if someone had a lottery ticket and said i took this ticket out of an unknown number of tickets with atleast one winning ticket in them, what is the chance it is winning. The coin could be bent in a way that could make the chance of flipping heads anywhere from 0-100% and in the ticket example the probabilty could either be 100% or in the range between aproximately 0-exactly 50%. I would have to say in both situations you dont have enough info to draw a guess on the probabilty

[wtfsvi]

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The same would be true if my bent coin was a two-headed coin. Are you suggesting a two-headed coin has probability 50% of coming up tails? Of course not. You are not really betting on the bent coin. You are betting on your fair coin.

[/ QUOTE ] A two headed coin is not the same scenario as the OP. A coin that has two identical sides, but we don't know if they are heads or tails, is the same as the OP. And in such a scenario, we can assume a 50% chance of heads if we flip the coin.

edit: I agree, however, that using a fair coin to pick the bets is not good in pedagogical terms. There is no correlation between the facts that the fair coin is 50/50, and that we can assume that the bent coin is 50/50. A 11/10 bet on the bent coin will be +EV no matter what method we use to pick heads/tails. And any event with two possible outcomes that we know nothing about the probability of, can be assumed to have a 50% chance of each outcome.

[ALawPoker]

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But suppose you walk into a bar and see the AA and KK on the table, with the rest of the deck in the center. Someone offers to take the KK side and bet against you. I hope you would not accept this bet without first demanding to inspect the rest of the deck and then have it thoroughly shuffled.

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If you offered me one of the sides, I'd of course assume that was the worse side if there was a chance you had more information than I did. Maybe David didn't set up your point correctly, in which case I'd correct him now, because it seems you are changing it slightly. In the coin example, the person who bent the coin does not pick a side. He lets you pick it, according to the OP.

It seems undeniable to me that I would still have a 50% chance of picking the winning side. In the AA vs. KK example, the edge of the AA side shrinks rapidly as you introduce an imperfect shuffle. Of course it does. But still, if I'm able to pick one of the sides, my odds should never get to worse than 50/50. If in doubt, I take a (unbent) coin out of my pocket and flip it. If the other guy chooses a side and I'm forced to take the one he's willing to give me, then that's a different situation, and of course one I would not like my odds in.

A friend of mine and I do this thing where we make a bet on a baseball game every week. One person picks the bet, and the other person chooses a side (and then we swap next week). If he picks something that's hard for me to judge which side has the edge, and/or I think he might be trying to deceive me into the wrong side with something that's deceptively attractive, I can always flip a coin and accept my 50/50 shot. It doesn't matter that I don't "know" whether or not Aaron Harang has a >50% chance of lasting more than 6 innings.

In the coin example there's no way to think I might have any insight at all into which side the coin will fall on. So I accept that, and pick at random. And I have a 50% chance of picking correctly.

[NotReady]

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There is more to be said but this will get things started.


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I don't see any problem with anything you said except where did you get the 11 to 10? If the bend is extremely slight then you could have a bankroll problem. And if the guy providing the coin knows it's bent and offers 11 to 10 I wouldn't go for less than 5 to 3. Otherwise "you're going to wind up with an ear full of cider".

[wtfsvi]

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I don't see any problem with anything you said except where did you get the 11 to 10? If the bend is extremely slight then you could have a bankroll problem. And if the guy providing the coin knows it's bent and offers 11 to 10 I wouldn't go for less than 5 to 3. Otherwise "you're going to wind up with an ear full of cider".

[/ QUOTE ] This does not make any sense. You're +EV at 11 to 10 if the coin is bent very slightly, very heavily, or not bent at all. It doesn't matter at all for your EV. You have a 50% chance of picking the right side and get paid 11 to 10.

[T50_Omaha8]

I think the only arguments in this thread that aren't trivial are the ones that are trying to show that the rest of the thread is trivial.

[jason1990]

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A coin that has two identical sides, but we don't know if they are heads or tails, is the same as the OP. And in such a scenario, we can assume a 50% chance of heads if we flip the coin.

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If I meet you in person, I will bring such a coin. I will tell you it has two identical sides, but I will not tell you they are both heads. Since you will assume a 50% chance of heads, you will be happy to accept my wager when I offer you 3 to 2 odds that it will come up tails. It sounds like free money for me.

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A 11/10 bet on the bent coin will be +EV no matter what method we use to pick heads/tails.

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I have a coin that I bent last year. It has come up heads 992,178 times in the last million flips. I would like to suggest a method for you to use to pick heads/tails. Go buy a lottery ticket. If you win, pick heads. If you lose, pick tails. I will offer you 11 to 10 odds on this bet. Do you think this is +EV for you?

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any event with two possible outcomes that we know nothing about the probability of, can be assumed to have a 50% chance of each outcome.

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This is exactly the kind of thinking my comments are meant to correct. Of course you can assume anything you want. That does not mean you are right. You assume this at your own peril. If you are going to assume that two things are equally likely, then you should have a good reason for assuming this, especially if you are going to act on that assumption in any significant way.

[wtfsvi]

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you will be happy to accept my wager when I offer you 3 to 2 odds that it will come up tails. It sounds like free money for me.

[/ QUOTE ] No. This is were your argument trips. The moment you offer me this I do know something about the probability of the two outcomes. Therefore I will not take your bet.