The Bent Coin

[NotReady]

[ QUOTE ]

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.


[/ QUOTE ]

It does matter because you base your future bets on the result of the first flip so you odds of picking the right side will depend on the amount of the bend.

[Phil153]

This whole thread come to down to DS et al saying "if you have only this information" and jason saying "but what if you have other information!"

Jason's only real point seems to be that probability models need to take into account all available evidence, and not just a portion of it. He should just say so because I don't think there's anyone here who disagrees. A far more interesting topic would be whether probability modelling (taking into account ALL the evidence) can be useful in some situations. Or whether people fail to use probability when it would yield useful results.

I think everyone here is intelligent enough to not need a cautionary tale of a two-headed coin. The strange thing is he thinks it's fine to use probability in the AA vs KK example, where you could just as easily posit a card mechanic as a two headed coin. The amusing thing is that he's used straight up probability in both cases anyway to make his point - the odds of being hustled in a coin wager vs the odds of getting a fair deal in a poker game.

[wtfsvi]

[ QUOTE ]

It does matter because you base your future bets on the result of the first flip so you odds of picking the right side will depend on the amount of the bend.

[/ QUOTE ] You don't base your future bets on the results of the first flip in the scenario in the OP.

[jason1990]

[ QUOTE ]
In the coin example, the person who bent the coin does not pick a side.

[/ QUOTE ]
He does. From the OP:

[ QUOTE ]
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 ]
I bent it. I picked the side. The side is heads.

My original point here is very simple. There is a bent coin. Consider these two events:

A1: the coin lands heads
A2: the coin lands tails

These events are not equally likely. Any reasonable person should admit that, and any reasonable person should tell you that you cannot determine the probabilities of these events without doing further analysis on the coin.

A few people, however, will tell you that they are equally likely. They use their lack of knowledge to assume they are equally likely, and then present that assumption as some sort of empirical fact. It is simple to see their error in the bent coin example. It is harder to see their error in more complicated examples.

David has shifted the focus away from A1 and A2. He has introduced:

B1: David's fair coin and my bent coin land on the same face.
B2: David's fair coin and my bent coin land on opposite faces.

These events are equally likely. Any reasonable person should admit this. And anyone with basic probability knowledge should be able to prove that this follows from the fairness of David's coin. If we have good, practical reasons to believe that David's coin is fair, then there is nothing to discuss. It is trivial.

David breaks even betting on B1 or B2, but this has nothing to do with my original point.

[NotReady]

[ QUOTE ]

You don't base your future bets on the results of the first flip in the scenario in the OP.


[/ QUOTE ]

From the OP:

[ QUOTE ]

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.


[/ QUOTE ]

[vhawk01]

[ QUOTE ]
[ QUOTE ]
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.

[/ QUOTE ]
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.

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

[/ QUOTE ]
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?

[ QUOTE ]
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.

[/ QUOTE ]
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.

[/ QUOTE ]
Your suggestion wasn't that you offer him 3:2 on a HEADS bet, it was that you offer him 3:2 and he can pick either heads or tails without knowing which it is. The fact that you know doesn't change unless you are the one picking. I really hope your entire point doesn't come down to this obvious misunderstanding.

[wtfsvi]

[ QUOTE ]
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.

[/ QUOTE ] Yes. Sklansky should not have introduced this "fair" coin since it is determined to cause misunderstandings. I'm not quite sure what your misunderstanding is, so it's hard for me to address, but I'm quite sure you misunderstand.

[jason1990]

The whole two-headed coin thing is completely tangential to my original point. My original point addresses people, like yourself, who say this:

[ QUOTE ]
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.

[/ QUOTE ]
Here is my original point, rephrased and located elsewhere in this thread:

--------------------

My original point here is very simple. There is a bent coin. Consider these two events:

A1: the coin lands heads
A2: the coin lands tails

These events are not equally likely. Any reasonable person should admit that, and any reasonable person should tell you that you cannot determine the probabilities of these events without doing further analysis on the coin.

A few people, however, will tell you that they are equally likely. They use their lack of knowledge to assume they are equally likely, and then present that assumption as some sort of empirical fact. It is simple to see their error in the bent coin example. It is harder to see their error in more complicated examples.

[NotReady]

[ QUOTE ]

I'm not quite sure what your misunderstanding is, so it's hard for me to address, but I'm quite sure you misunderstand.


[/ QUOTE ]

I should save this kind of frustration for my posts on theism - at least then I feel the aggravation has a good purpose.

[jason1990]

[ QUOTE ]
Jason's only real point seems to be that probability models need to take into account all available evidence

[/ QUOTE ]
This is not my point at all. I know my posts are esoteric, but keep reading. You should be able to understand my latest ones.