2nd Flip of a bent coin

[luckyme]

I have a friend bend a coin as much as he wants out of my sight. We agree on an even money bet on the 1st flip. I randomize my choice 50-50.

It flips heads, 3rd party judge reports to us blind.

I want to do another even money bet on the 2nd flip. I'm +/=EV taking heads this time. The range could be from 0 to 1. Is my blind start and 50-50 assumption now irrelevant or am I actually building on it?

If a different unknown coin is flipped 6 times, all heads, I'm tempted to say "hey, let's see that bloody coin!" Where does that come from, other than my theoretical 50-50 expectation?

I'm starting to think there is a blending of two concepts going on both involving 50-50. One is the bet-protection that the first either-or situation causes. The other is the theoretical expectation of a fair coin which we compare a string of results to.

Is it a different 50-50 that is being referred to in those cases? the 1st flip one and the repeated test one. That's what it looks like to me but I'm a Sklamorian.

luckyme

[PairTheBoard]

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I have a friend bend a coin as much as he wants out of my sight. We agree on an even money bet on the 1st flip. I randomize my choice 50-50.


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So you're betting with your friend who bent the coin? Is that the way you intended to set it up? If it is then it's a good thing the agreed to bet allows you to call the flip after the bet's been made. You are really betting on that fair coin you are flipping to call the bet. You are betting it will flip to a match.

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It flips heads, 3rd party judge reports to us blind.

I want to do another even money bet on the 2nd flip. I'm +/=EV taking heads this time.

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Not if you're betting with your friend and he knows how he bent the coin and he agrees to the bet. Chances are he has folded it away from tails to produce a tails bias. Since it just happened to land heads first flip he gets a nice +EV for himself when he agrees to your second bet.

So maybe you didn't intend such a complication. Let's assume you are gambling with someone who knows nothing more about the Bent Coin than you do. I can't see any way it could be argued that Betting Heads at Even Money is not +EV, assuming the Bent Coin has any bias at all. The Big Question is, what are the worst odds you would take to bet Heads? Your randomizing Call of the coin on the first bet really begs the question of how you view the No-Information condition for the First Flip in terms of Probalities involving the Bent Coin. You could know the Bent Coin has a Tails bias and your randomizing Call would work just as well. Your even money on the First Bet is the payoff on P(Fair Coin flips to a Match).

What are the worst odds you would take to bet Heads after the First Flip? That will tell us how you are incorporating your No-Information into your Known Information about the First Flip.

PairTheBoard

[luckyme]

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What are the worst odds you would take to bet Heads after the First Flip? That will tell us how you are incorporating your No-Information into your Known Information about the First Flip.


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Perhaps my friends may operate at a different level than most friends, some have signing authority on my chequing account. So, let's leave the 'he's trying to screw you' part aside and just say he's in an investigative mode like I am.

The question I'm poking around is whether the pre-first flip 50-50 bet, which is a game-theory protection approach to it, has any connection to the 50-50 expectation that I would base my wanting to bet the way the coin flips the 1st time. Actually, I don't think it matters if we know the coin is bent or not, if I'm dealing with a unknown 'normal' coin and it flips heads the first time I'd want to bet even money on heads the second time.

My view is the 50-50 that I refer to for that 1st bent coin flip is a different 50-50 than the one that makes me want to bet even money on heads on any coin if the 1st flip is heads. Trying to put a number on my EV by doing that is on a different topic, I'm focusing on the 'nature' of those two 50-50's.

thanks for looking at this, btw, luckyme

[PairTheBoard]

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The question I'm poking around is whether the pre-first flip 50-50 bet, which is a game-theory protection approach to it, has any connection to the 50-50 expectation that I would base my wanting to bet the way the coin flips the 1st time. Actually, I don't think it matters if we know the coin is bent or not, if I'm dealing with a unknown 'normal' coin and it flips heads the first time I'd want to bet even money on heads the second time.


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You are confusing me. On the Second Flip, Do you know the coin has been bent or not? You say you would want to bet even money on Heads. Would you be just as happy to bet even money on Tails? If not, why not?

PairTheBoard

[luckyme]

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The question I'm poking around is whether the pre-first flip 50-50 bet, which is a game-theory protection approach to it, has any connection to the 50-50 expectation that I would base my wanting to bet the way the coin flips the 1st time. Actually, I don't think it matters if we know the coin is bent or not, if I'm dealing with a unknown 'normal' coin and it flips heads the first time I'd want to bet even money on heads the second time.


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You are confusing me. On the Second Flip, Do you know the coin has been bent or not? You say you would want to bet even money on Heads. Would you be just as happy to bet even money on Tails? If not, why not?

PairTheBoard

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Sorry for any confusion, I'm trying to sort out the concept(s) of 50-50.

Bent case - I know it is bent by an unknown amount in an unknown direction flipped heads on the 1st toss.

Random new coin case - a supposedly normal coin just flipped heads for the 1st time.

1st question, a math one - I want to bet even money ( setting all 'he's out to getcha' worries aside) that the coin will land heads the 2nd time. In both the bent and unbent cases. Is that sound?

2nd question, and the real point of my post - is the 50-50 concept I used to protect myself in the first flip of the bent coin really the same 50-50 concept that I'm using when I want to bet even money on heads on the 2nd flip of the fair coin.

Perhaps I'm stumbling with this because the bent coin pre 1st flip I'm not dealing with reality and merely using a strategy that protects my position. No claim is made that there is an actual 50-50 chance in results.

In the 2nd bet cases, I have a test result, and essentially applying some version of the 'burned by a hot stove' reasoning. Iow, once an actual physical system is in place and knowing I can't possible have all the info involved, and knowing that a physical system is never perfect the specific result in this expected 50-50 split seems to be an different 50-50 concept.

In the bent coin one 50-50 never was thought of as a property of the coin. In the 'normal' coin, 50-50 is actually being treated as a baseline expectation.

One 50-50 seems to deal with actual properties of the coin ( let that terminology slide) and the other doesn't deal with them at all.

thanks, luckyme

[jason1990]

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I have a friend bend a coin as much as he wants out of my sight. We agree on an even money bet on the 1st flip. I randomize my choice 50-50.

It flips heads, 3rd party judge reports to us blind.

I want to do another even money bet on the 2nd flip. I'm +/=EV taking heads this time. The range could be from 0 to 1. Is my blind start and 50-50 assumption now irrelevant or am I actually building on it?

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You are actually building on your 50-50 assumption. The math is a little technical. f97tosc talked about "priors" here:

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Suppose that we write f(P) for the prior probability distribution that the coin is such that when tossed, it will result in heads a long-term fraction P of the time.

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The function f(P) is the prior. Essentially, P is the probability of heads for the bent coin, and f(P) is your estimate of the probability that the coin is actually bent that way. For example, if you think all bends are equally likely, then the function f(P) is flat. If you think extreme bends are less likely than small bends, then the function f(P) has a hump in the middle.

Your initial 50-50 assumption is represented by choosing a prior with a certain property. For example, if you choose a prior which is symmetric about the line P = 1/2, then you would get a 50-50 initial estimate. The actual shape of the prior beyond that is up to you.

You must use the prior if you want to apply Bayes theorem and estimate the probability of heads on the second flip. This estimate will depend on the exact shape of the prior, and not simply on its symmetry. So it depends on more than just your initial 50-50 assumption. But you can say, based only on symmetry, that the second flip is more likely to be heads than tails. As f97tosc says in his post,

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we only need to make some very mild additional technical assumptions to be able to conclude that, for example, after seeing one head, we rationally should judge it more likely than not for the next toss to be head as well

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This means that if you start with a prior function f(P) that gives you a 50-50 initial estimate, then the fact that heads is more likely than tails on the second flip does not depend on the shape of the function. Even though the exact probability of heads on the second flip does depend on the shape of the function.

In the end, though, the conclusion that heads is more likely on the second flip is based both on the first flip being heads and on your initial 50-50 assumption.

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If a different unknown coin is flipped 6 times, all heads, I'm tempted to say "hey, let's see that bloody coin!" Where does that come from, other than my theoretical 50-50 expectation?

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It is a mix of both your initial assumption and the 6 flips. You might say it is "mostly" the 6 flips. The reason you might say that is because, even if your initial assumption was biased towards heads, 6 heads is pretty hard to get. So as long as the bias in your assumption was not too extreme, you would still be surprised by 6 heads in a row.

But it also comes from another assumption you might be making. You might imagine that extreme bends are less likely than small bends. That would mean your prior function f(P) has a hump in the middle. In some sense, you must make this assumption in order to be surprised by 6 heads. For example, a flat prior would make the probability of 6 heads exactly 1/7. In fact, it makes the probability of k heads exactly 1/7 for all k. If you initially assume that all bends are equally likely, then all numbers of heads are also equally likely, and you should not be surprised if you see 6 heads or 3 heads or 0 heads. They would all have the same chance of happening, if all bends were equally likely.