The Bent Coin

David Sklansky · #1 06-02-2007, 02:38 PM

I don't think this discussion will have legs because I don't think jason's esoteric comments about probability interest many people here. But I do want to give a precise reply.

Jason says that if I tell you I bent a coin but don't show it to you, you cannot state the probability as to whether a flip will come up heads. If asked, the answer is "I don't have enough information."

I say that the information you have, in this case only the fact that there is two alternatives, allows you to break even on your bets if you flip a "fair" coin, use the the result of that flip to choose a side for the bent coin and get even money on your bets. You would win getting eleven to ten.

Any other definition of probability seems silly. Because you NEVER in real life have enough information. There is no fair coin.

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

chezlaw · #2 06-02-2007, 02:54 PM

Don't we have to be careful what we are talking about the probability of?

If you get to call then you have a 50:50 chance of winning whether or not the coin is bent but that's not because the coin has a 50% chance of landing on heads or tails which is why you wouldn't let the guy who knows the coin call.

The reason you still have a fair chance if you call is because there's no disadvantage in not knowing which way the coin is unfair even if you know its unfair.

chez

Phil153 · #3 06-02-2007, 03:13 PM

I don't understand the significance or relevance of Jason's coin.

Probability is about using the information you have available to you. If there is unknowable information, even if it changes your chance of winning to 100% or 0%, it in no way diminishes the probability model.

If he's trying to show that probability fails when there is non quantifiable information, a bent coin analogy isn't going to help.

Am I missing something?

David Sklansky · #4 06-02-2007, 03:17 PM

I should say at his point that I am not sure I have described Jason's contention correctly.

ALawPoker · #5 06-02-2007, 03:20 PM

I don't understand this either. Jason's claim seems absurdly semantical. 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). But you have enough information to make the best decision available, which is all that matters.

chezlaw · #6 06-02-2007, 03:28 PM

[ QUOTE ]
I don't understand this either. Jason's claim seems absurdly semantical. 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). But you have enough information to make the best decision available, which is all that matters.

[/ QUOTE ]
What's the probability of the zillionth digit of Pi being a three?

chez

Phil153 · #7 06-02-2007, 03:41 PM

[ QUOTE ]
I should say at his point that I am not sure I have described Jason's contention correctly.

[/ QUOTE ]
I've finally read his post (I avoid esoteric stuff, prefer it easy) and his points are not clear, and where clear, they are not correct.

The only real contention to your OP is that the real world is never modeled by your conditions. Or that where it is, it is impossible or extremely difficult to combine probability and non probability considerations.

Which is a very interesting debate that we should be having instead.

David Sklansky · #8 06-02-2007, 03:49 PM

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?

m_the0ry · #9 06-02-2007, 03:58 PM

[ QUOTE ]
What's the probability of the zillionth digit of Pi being a three?

chez

[/ QUOTE ]

Depends on the numerical base, not enough information.

[img]/images/graemlins/tongue.gif[/img]

jason1990 · #10 06-02-2007, 04:43 PM

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Jason says that if I tell you I bent a coin but don't show it to you, you cannot state the probability as to whether a flip will come up heads.

[/ QUOTE ]
Based on the rest of your reply, what is your answer? 50%?

[ QUOTE ]
I say that the information you have, in this case only the fact that there is two alternatives, allows you to break even on your bets if you flip a "fair" coin, use the the result of that flip to choose a side for the bent coin and get even money on your bets.

[/ QUOTE ]
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.

The same would be true of anything you might bet on, in fact. You could flip your fair coin, if it comes heads you bet that the next roll on the craps table will be a 7, if it comes tails you bet that it will not be a 7. If someone pays you even money on your bets, you will break even. Do you really think this says something about the probability of rolling a 7? Of course not. You are not really betting on the roll of the dice. You are again betting on your fair coin.

The challenge is to

[ QUOTE ]
state the probability as to whether a flip will come up heads.

[/ QUOTE ]
You evade the question by using your fair coin to sometimes bet on tails. If you had to bet only on heads, what odds would you require? That is the question.

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you NEVER in real life have enough information. There is no fair coin.

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Are you simply trying to point out that no coin is perfectly fair? If so, then I must say I do not find that topic interesting right now. My comments have so far been meant to address practical conclusions in ordinary circumstances. I have no present interest in discussing coin-flipping machines, coins that land on their edge, or the fact that the heads side of a quarter might be heavier than the tails side. Those discussions have their place, but they are not what I am talking about here.

A practical statistician will tell you that, under ordinary circumstances, it is reasonable to conclude that a typical unbent coin will land heads with probability 1/2. That same practical statistician will tell you that you cannot determine the probability of heads in a bent coin without further information. He would probably tell you that the best way to determine it is to flip the coin many times and use the results of those flips to estimate this probability.

This really is a very trivial observation on my part. I am astounded that you are so intent on debunking it.