The Mysterious Coin
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The Mysterious Coin
Suppose you have a standard coin with a head on one side and a tail on the other. However, you don't know the probability the coin will land on heads  it could be anywhere from 0 to 1. Because any probability is equally likely, the overall probability that you will get a heads when tossing the coin is 1/2. Now, suppose you toss the coin once and it lands on heads. What is the probability that it will land heads if you toss it again?
Re: The Mysterious Coin
P(HeadHead)= .50; coin tosses are independent events, meaning that the outcome of one coin toss (event) does not affect the outcome of another.
Re: The Mysterious Coin
I may be wrong, but I think that's incorrect, and here's why:
Spoiler:
Re: The Mysterious Coin
gaga654 wrote:Suppose you have a standard coin with a head on one side and a tail on the other. However, you don't know the probability the coin will land on heads  it could be anywhere from 0 to 1. Because any probability is equally likely, the overall probability that you will get a heads when tossing the coin is 1/2. Now, suppose you toss the coin once and it lands on heads. What is the probability that it will land heads if you toss it again?
I bolded the part that’s false.
But assuming it were true, the answer would be
Spoiler:
On the other hand, assuming the much more likely situation that, when I first encounter “a standard coin” my prior probability for it is a sharp spike around 1/2, then futurityverb is essentially correct. At least, each individual datum has far less of an impact on my prior distribution for it.
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Re: The Mysterious Coin
Yay, Bayes' theorem...
But yes, that's taking the uniform prior as a given... in real life, with a real coin, that's not a reasonable prior, so you'd have a different result.
Spoiler:
But yes, that's taking the uniform prior as a given... in real life, with a real coin, that's not a reasonable prior, so you'd have a different result.
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void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
Re: The Mysterious Coin
I realize that in real life it would not be equally likely to have any probability, but for the sake of this problem assume that it is. I don't know what the actual answer is, but I believe that phlip and Qaanol are correct.
Re: The Mysterious Coin
Followup question:
Again starting from a uniform prior, how many times in a row must the coin land on heads (without ever landing on tails) before you are 95% confident that the coin is not fair (ie. that the true odds are not 5050)?
Again starting from a uniform prior, how many times in a row must the coin land on heads (without ever landing on tails) before you are 95% confident that the coin is not fair (ie. that the true odds are not 5050)?
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 phlip
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Re: The Mysterious Coin
Qaanol wrote:Followup question:
Again starting from a uniform prior, how many times in a row must the coin land on heads (without ever landing on tails) before you are 95% confident that the coin is not fair (ie. that the true odds are not 5050)?
169 answer:
Spoiler:
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enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
Re: The Mysterious Coin
phlip wrote:Qaanol wrote:Followup question:
Again starting from a uniform prior, how many times in a row must the coin land on heads (without ever landing on tails) before you are 95% confident that the coin is not fair (ie. that the true odds are not 5050)?
169 answer:Spoiler:
Ahem. How many times must the coin land on heads without landing on tails before you are 95% confident that the coin is biased in favor of heads?
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Re: The Mysterious Coin
Qaanol wrote:How many times must the coin land on heads without landing on tails before you are 95% confident that the coin is biased in favor of heads?
Spoiler:

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Re: The Mysterious Coin
As a side note, it's bad statistical procedure to generate a confidence interval without at least 510 of both observations.
Re: The Mysterious Coin
ARandomDude wrote:As a side note, it's bad statistical procedure to generate a confidence interval without at least 510 of both observations.
Nah, it's okay. This is only a problem if you are relying on (antiquated) approximations.
Re: The Mysterious Coin
There is always a variable forgotten in the coin flip questions. And that is how the coin is flipped.
I was bored one month in my teens and practiced flipping coins. A standard USA nickel makes a perfect flipper if you want to destroy any probabilities (or win bets), surprisingly better than the quarter, at least for me. Learn to flip uniformly and you can flip/predict with near 100% accuracy. Just like shooting free throws, all it takes is practice. I believe I stopped after 28 tails in a row once I learned to flip, and I could predict more than that. Then I was bored with that, moved on.
I was bored one month in my teens and practiced flipping coins. A standard USA nickel makes a perfect flipper if you want to destroy any probabilities (or win bets), surprisingly better than the quarter, at least for me. Learn to flip uniformly and you can flip/predict with near 100% accuracy. Just like shooting free throws, all it takes is practice. I believe I stopped after 28 tails in a row once I learned to flip, and I could predict more than that. Then I was bored with that, moved on.
Re: The Mysterious Coin
Spiky wrote:Learn to flip uniformly and you can flip/predict with near 100% accuracy.
That has got to be the most useful skill EVER. Holy crap.
Adam
Re: The Mysterious Coin
Adam H wrote:Spiky wrote:Learn to flip uniformly and you can flip/predict with near 100% accuracy.
That has got to be the most useful skill EVER. Holy crap.
Here's a video of me doing just this. If I had no skill, I would have expected to need about 512 recording attempts to get ten heads in a row, but I think this was the third or fourth try.
Re: The Mysterious Coin
It's also REALLY easy if you're allowed to catch it in one hand then put it on the back of your other. The reason being that it's relatively easy to run your thumb against the side of a quarter and tell if you're rubbing either heads or tails. Then it's just a matter of flipping it without being caught which is pretty easy to do as well.
If you're having somebody else call it and you want to make it land opposite of what they call it doesn't help you much if the other person calls it after you flip it and put it on your wrist so you really do need them to "call it in the air".
If you're having somebody else call it and you want to make it land opposite of what they call it doesn't help you much if the other person calls it after you flip it and put it on your wrist so you really do need them to "call it in the air".
double epsilon = .0000001;

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Re: The Mysterious Coin
I have a problem, assuming uniform prior you flip the coin and it comes up heads three times and tails once, what is the probability that on the next toss it will come up heads?
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
Re: The Mysterious Coin
tomtom2357 wrote:I have a problem, assuming uniform prior you flip the coin and it comes up heads three times and tails once, what is the probability that on the next toss it will come up heads?
If you flip a+b times and get a heads and b tails the probability of a head on the next go is
Spoiler:
which for a=3, b=1 gives
Spoiler:

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Re: The Mysterious Coin
Can you solve those integrals please?
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Re: The Mysterious Coin
tomtom2357 wrote:Can you solve those integrals please?
Spoiler:

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Re: The Mysterious Coin
I mean a general solution.
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 phlip
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Re: The Mysterious Coin
I'm sure you could write out a general form... they're just integrals of polynomials, after all. But you'd have to expand the binomial out, and you'd end up with something much uglier and less usable than the original integral form.
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enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
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Re: The Mysterious Coin
Okay, I guess that I can work with the integral. Also, I have an interesting problem, I have a strange coin, in that for the first coin toss, is is 5050 to land on heads or tails, but on every toss after that, the coin has a 75% chance of landing on the face it landed on last time. What is the distribution of the tosses now (e.x. a normal coin has a binomial distribution, which tends to a normal distribution as the amount of tosses approaches infinity)
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
Re: The Mysterious Coin
It still converges to the binomial distribution (but with a different variance) and to the normal distribution (again with a different variance), but you see larger deviations from the distribution for small numbers of coin tosses.
To calculate, you can describe the system with two distributions together: one for "last toss was head" and one for "last toss was tail".
To calculate, you can describe the system with two distributions together: one for "last toss was head" and one for "last toss was tail".

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Re: The Mysterious Coin
Are you sure? When I tried it for small numbers of tosses, it looked like there was a sudden dip in the distribution at about the center.
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Re: The Mysterious Coin
That in no way precludes eventual *convergence* to distributions without such a dip.tomtom2357 wrote:Are you sure? When I tried it for small numbers of tosses, it looked like there was a sudden dip in the distribution at about the center.

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Re: The Mysterious Coin
Okay then, could you graph the distribution then and show me?
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 gmalivuk
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Re: The Mysterious Coin
Try doing it yourself. You'll learn more than if you constantly expect everyone else to prove things for you.
Re: The Mysterious Coin
phlip wrote:I'm sure you could write out a general form... they're just integrals of polynomials, after all. But you'd have to expand the binomial out, and you'd end up with something much uglier and less usable than the original integral form.
Actually I think you can get a nice form in terms of factorials if you restrict a and b to be integers.
double epsilon = .0000001;
Re: The Mysterious Coin
 Indigo is a lie.
Which idiot decided that websites can't go within 4cm of the edge of the screen?
There should be a null word, for the question "Is anybody there?" and to see if microphones are on.

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Re: The Mysterious Coin
I think that macbi just killed the thread.
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Re: The Mysterious Coin
Wait, so the actual solution is a/b, or a+1/b+2?
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 jestingrabbit
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Re: The Mysterious Coin
tomtom2357 wrote:Wait, so the actual solution is a/b, or a+1/b+2?
Its right there in the wikipedia article. Read it, and read your private messages by accessing your user control panel, linked towards the top of the page.
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Re: The Mysterious Coin
Supose you have got an special skill to detect a defective coin before the first flip.
You make a prior, probably a 6040 distribution in favour to heads.
You start testing
As you are a bayesian man, you rely on your prior , a 6040 frequency in your next 200 tosses might mean money to you
A frequestist needs more data
How much data takes to know the accurate frequency?
You make a prior, probably a 6040 distribution in favour to heads.
You start testing
As you are a bayesian man, you rely on your prior , a 6040 frequency in your next 200 tosses might mean money to you
A frequestist needs more data
How much data takes to know the accurate frequency?
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