Calling All Mathmeticians and Statisticians

[David Sklansky]

Please go over to the SMP forum for a minute and weigh in on the Bent Coin and Clarifying The Bent Coin threads.

[f97tosc]

It looks like the "Bent Coin" thread is not the first thread on the matter. Where is the original problem formulation?

[David Sklansky]

Its kind of complicated. I don't think you need to read the earlier threads.

[SamIAm]

I assume you mean this one. (Linking to save others the search.)
-Sam

edit: I found the original thread to be way worthwhile. Jason makes a lot of points in that thread that were ignored in the second.

[djames]

Sam, your sentence about Jasons' ignored points is crucial. Unfortunately, it is likely the true reason why Sklansky started the second bent coin thread, and wrote the post just above yours within this one. Four attempts (as of this post) by Jason requesting a lucid & direct response to his simple questions from Sklansky have gone unanswered. If a response ever does come, it surely will sidestep the question somehow. Probably by posing another murky example we'll have to wade through anew.

A bit more of the history should uncover 3 to 5 prior threads that have evolved into the 2 bent coin threads. (Someone better with Search could surely post the chronography for interested readeres). Within these, PairTheBoard has been the primary antogonist to Sklansky, and in my opinion, has faired wonderfully. Sklansky does have a big flock though...

[Beavis68]

I am not sure if anyone has done the math, but I think this is it.

Chance of the bent coin coming up heads = x chance of the bent coin coming up tails = 1-x.

Chance of you flipping heads with your coin when the bent coin is heads = 1/2 odds of flipping tails when the bent coin comes up tails = 1/2

The probability of the bent coin being heads and you flipping heads is x*1/2=1/2x

Probability of the bent coin coming up tails and you fillping tails on your coin = (1-x)*1/2 = 1/2 - 1/2x.

the probablility you will flip the correctly and win = 1/2x + (1/2 - 1/2x) = 1/2

What I got my brain around that made me accept this is that when you flip whatever why the coin is baised too, you will be correct much more often, so this even it out.

the bias (x) is not relavant to the equation

[CT11]

I'm reluctant to respond in this thread but the other threads are so incoherent I can't even tell what the argument is about.

If the question is what I think it is Beavis68 seems to be on the right track though his argument is a little informal.

Can I assume the game is this:
I may choose heads or tails using whatever strategy I feel like. You will flip a coin where P(heads) =p and P(tails)=1-p. I win k dollars if I guess the same as the coin flip and I loose c dollars if I guess the opposite. One of these outcomes will happen almost surly.

Is that the game?

ok whats the question?

is this the question:
you have a choice between playing two games, A and B.
game A:
as above k=11, c=10, p=0.5. you play the game.
game B:
as above k=12, c=10, p is fixed and unknown and an element of [0,1]. you play the game.

which game has a higher expected value if you use the optimal strategy for each one?

edit: made it a bit more clear

[SamIAm]

[ QUOTE ]
I am not sure if anyone has done the math, but I think this is it.

Chance of the bent coin coming up heads = x chance of the bent coin coming up tails = 1-x.

Chance of you flipping heads with your coin when the bent coin is heads = 1/2 odds of flipping tails when the bent coin comes up tails = 1/2

The probability of the bent coin being heads and you flipping heads is x*1/2=1/2x

Probability of the bent coin coming up tails and you fillping tails on your coin = (1-x)*1/2 = 1/2 - 1/2x.

the probablility you will flip the correctly and win = 1/2x + (1/2 - 1/2x) = 1/2

What I got my brain around that made me accept this is that when you flip whatever why the coin is baised too, you will be correct much more often, so this even it out.

the bias (x) is not relavant to the equation

[/ QUOTE ]
I'm not sure what your point is. You're right that using your own coin turns the bent-payoff into an unbent-payoff. But of course it's a red herring. (I'm hoping that's your point with the irrelevance of x.) This is one of the points Jason made in the earlier thread that's being ignored by NewThreaders.

[ QUOTE ]
Sam, your sentence about Jasons' ignored points is crucial. Unfortunately, it is likely the true reason why Sklansky started the second bent coin thread

[/ QUOTE ]
I'd hope that it wasn't his conscious effort, but I agree that it had that effect. [img]/images/graemlins/frown.gif[/img]

[ QUOTE ]
I'm reluctant to respond in this thread but the other threads are so incoherent I can't even tell what the argument is about.

[/ QUOTE ]
I'd say that the question is NOT "how do you make money from the situation?" Everybody agrees that the unbent coin makes the problem trivial. The question that they do NOT agree on is "What is the probability that the coin lands on Heads?".
-Sam

P.S. I think all the talk about Persi as this Math Oracle is pretty awesome. I've hung-out with the guy at conferences; we both study MarkovChains and random sampling and he does great work. He's even my advisor's advisor. But I'm not sure how he became the Probabalistic Truthsayer in that thread. [img]/images/graemlins/smile.gif[/img]

[SamIAm]

Doh. I just saw Beavis's x-post in the SMP discussion. Bruce, would you mind locking this one, so we don't have competing threads?

[Beavis68]

"This is one of the points Jason made in the earlier thread that's being ignored by NewThreaders."

Can you link to this? I from what I have read Jason has a mental block and this example is not a red herring. It doesnt matter what method you use to guess heads or tails, you still have a 50/50 chance.

The point of saying that X is irrelavent is to show that it does not matter how the coin in biased or how biased it is to judge if the bet is a good one or not.