Coin tossing, probability and some logic, kinda

[Swan Red]

what are the odds it lands tails next time?

 

Most seem to argue that the probability of the coin landing tails is unconditional on previous coin tosses. I think this is a mistake, I've mentioned this before and Paul thinks it's silly and John has also referred to it.

 

My point is that the probability of it landing tails is dependent on the assumption that the coin is a fair coin. A fair coin being defined as one that is as likely to land heads as tails. What confidence should we have that this assumption is correct? It seems to me entirely feasible that some very small % of coins may though accident or design have some tendency towards heads or tails. If this is correct then we should consider the probability conditional.

 

If we toss a coin 10 times and it lands heads it seems we can't do worse by backing heads last time, if indeed some coins are biased then we should include that in our calculations and weight the probability accordingly.

 

We don't need 10 tosses for this though, if we employ Bayes Theorem we should our assessment of the probability should be modified every time we know something new, like how the coin landed last time.

 

Consider then the prior probabilities of two coin tosses, the standard response would be that the Probability of the 4 possible outcomes HH = HT = TH = TT = .25. The Bayesian disagrees, the Bayesian considers the following P(H|H) = P(T|T) > P(H|T) = P(T|H)

 

Why is the Bayesian wrong?

[surf]

i'm on board

[John am Rhein]

I think there's two kinds of problem here:

 

1) the statement is "toss a FAIR coin 10 times" - this means that the probability of the coin falling heads is 0.5, regardless of any previous outcomes - the bayesian calculation above would be wrong or, more specifically, inappropriate in that situation

2) the statement (as above) is simply "toss a coin 10 times" - nothing is said about whether the coin is fair or not. In this case you're using the results of previous coin tosses to get more information about how fair or unfair the coin is - the more previous tosses you include, the more information you have. The bayesian calculation probably makes more sense here that the default assumption that the coin is fair, but it's not necessarily the best way to estimate the bias of the coin - there's probably a more specific statistical test for that (I'll try to find out more because I can't remember the details)

 

10 consecutive heads is a bit of an extreme example, though - it's hard to imagine any kind of realistic-looking coin which would have such a bias unless it was really obvious to the eye, such as having heads on both sides, etc.

[charlie clown]

Head I win, tails you lose.

[RP]

That it's "a fair coin" is a given.

 

If it's not, then there can be no logical debate.

[Swan Red]

That it's "a fair coin" is a given.

 

If it's not, then there can be no logical debate.

 

We can have a logical debate if the status of the coin is undetermined.

 

More accurately how do we determine that it's a fair coin?

[RP]

Ok, fair point - but it would change the nature of the discussion significantly.

 

My previous points when we have discussed this still stand based on a fair coin.

 

With a coin of indeterminate bias, the increased number of repeated (i.e. same) results would start to introduce doubt as to fairness and therefore skew odds on futures tosses. This doubt would start, at a miniscule base, from just one toss - first result is a head, there has to be some (miniscule) doubt whether the coin is fair. That increases with every consecutive incidence of a head.

[Redwire]

The bayesian calculation probably makes more sense here that the default assumption that the coin is fair, but it's not necessarily the best way to estimate the bias of the coin - there's probably a more specific statistical test for that

 

Calculate the posterior probability density function.

 

[Swan Red]

Ok, fair point - but it would change the nature of the discussion significantly.

 

My previous points when we have discussed this still stand based on a fair coin.

 

With a coin of indeterminate bias, the increased number of repeated (i.e. same) results would start to introduce doubt as to fairness and therefore skew odds on futures tosses. This doubt would start, at a miniscule base, from just one toss - first result is a head, there has to be some (miniscule) doubt whether the coin is fair. That increases with every consecutive incidence of a head.

 

The assumption that it's a fair coin is the one I'm challenging. in fact I am claiming that all coins prior to tossing are of indeterminate bias, that the frequency of biased coins is extremely small is an acceptable assumption but once we accept that assumption the status of all coins becomes indeterminate.

 

I'm still only trying to get my head round this.

 

I think there's two kinds of problem here:

 

1) the statement is "toss a FAIR coin 10 times" - this means that the probability of the coin falling heads is 0.5, regardless of any previous outcomes - the bayesian calculation above would be wrong or, more specifically, inappropriate in that situation

2) the statement (as above) is simply "toss a coin 10 times" - nothing is said about whether the coin is fair or not. In this case you're using the results of previous coin tosses to get more information about how fair or unfair the coin is - the more previous tosses you include, the more information you have. The bayesian calculation probably makes more sense here that the default assumption that the coin is fair, but it's not necessarily the best way to estimate the bias of the coin - there's probably a more specific statistical test for that (I'll try to find out more because I can't remember the details)

 

10 consecutive heads is a bit of an extreme example, though - it's hard to imagine any kind of realistic-looking coin which would have such a bias unless it was really obvious to the eye, such as having heads on both sides, etc.

 

I agree with this but, and I should have clarified this the assumption that it's a fair coin is the one being challenged. And thanks I'd be interested in the more specific test and I'll look into Redwire's answer.

[John am Rhein]

Calculate the posterior probability density function.

 

:)/>/>

 

can't be arsed ;-)

[PhilM]

There's the issue of whether it's a fair coin, and whether it's a fair toss. If each toss was equal in terms of force and height, with a fair coin, you should (given that the coin's starting position would have to be the same) get either the exact same result each time, or alternating opposite results.

 

If you're not consistent with your toss, then surely the outcome is random based on how the coin is tossed, since that will have a far greater influence on the side of the coin which is facing up than any minute difference in weighting between the faces of the coin?

[£440,000]

If the coin is 100% fair, then wouldn't the Bayesian model work as the extra factor affecting the probability would be the absolute fairness of the coin. That would be the extra information. I don't think that the previous toss alone can ever be taken into account though. The previous 100 or 1000 tosses maybe.

 

If the coin is absolutely fair, the eleventh toss will always have odds of 50/50. The probability of it landing head eleven times in a row are tiny, but the probability of an individual toss is different from the probability of a sequence of like tosses.

 

The only factor that could change the probability is the nature of the coin and it's propensity to land on one side or the other.

 

Hope I don't sound like a tosser!

[John am Rhein]

This seems to be an excellent summary of the topic:

 

en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair

 

Just like Redwire said, the "Posterior probability density function" applies here and this relates to the bayesian approach you already mentioned. Then it gives the "estimator of true probability" which, as far as I can tell, just boils down to "divide the number of heads by the total number of coin tosses" so, for example, if you had 9 heads from 10 tosses, you'd estimate the probability of heads on a single toss to be 0.9.

[dorgie]

Surely the hand that tosses it has far more bearing on the likely outcome than whether it's a fair coin or not ? Is it not almost impossible for that hand to toss the coin with the same velocity, speed, height etc every time ?

[D.Boon]

At 7.00 in the morning?

[Earl Hafler]

What side of the coin is facing up when it's tossed ? Is it caught or left to fall on the floor ?

 

Haven't read the theorem link but have heard it before.

[Nebraska Red]

zzzzzzzzzzzzzzzzzzzzz

[Swan Red]

I don't understand NR is the Z corresponding to a side of the coin? In any case 20 would give me even more reason to consider the coin biased.

[surf]

I don't understand NR is the Z corresponding to a side of the coin? In any case 20 would give me even more reason to consider the coin biased.

 

maybe the cold weather is affecting the metal in non-symmetrical fashion

[John am Rhein]

I don't understand NR is the Z corresponding to a side of the coin? In any case 20 would give me even more reason to consider the coin biased.

 

z covers the case where the number of heads is a complex number, i.e. z = x + iy where i is the square root of minus 1

 

it's an application of quantum mechanics to the tossing question

[PhilM]

z covers the case where the number of heads is a complex number, i.e. z = x + iy where i is the square root of minus 1

 

it's an application of quantum mechanics to the tossing question

 

Or perhaps he's making claims about how the depth of the coin along the third axis will affect the outcome?

[JRC]

I would always prefer to work on the assumption that any toss was Platonic.

[John am Rhein]

This seems to be an excellent summary of the topic:

 

en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair

 

i've read the wikipedia article properly now and the specific bayesian example seems to rest on the rather dubious assumption that the 'prior distribution' of the actual probability of the coin showing heads on any single toss is a 'uniform distribution', i.e. just as likely to be 0.1 or 0.9 as it is to be 0.5 with all numbers between 0 and 1 being equally likely. Indeed the article itself says "it would be more appropriate to assume a prior distribution which is much more heavily weighted in the region around 0.5, to reflect our experience with real coins".

 

So the subtleties here are concerned with what should realistically be assumed about the fairness of the coin. In the real world, perfectly fair coins (and throws) are probably rare or even non-existent, but the likelihood of an unfair coin having a probability of heads that's significantly far from 0.5 is presumably pretty low as well. There's then the small, but significant, likelihood of a coin being 2-headed or 2-tailed which is presumably much more likely than a coin that somehow has a head-face and a tail-face but is somehow almost certain to land on one or the other.

[StevieC]

z covers the case where the number of heads is a complex number, i.e. z = x + iy where i is the square root of minus 1

 

it's an application of quantum mechanics to the tossing question

 

 

In this scenario, the result is both heads and tails. Until you check.

[DanielS]

In this scenario, the result is both heads and tails. Until you check.

 

get the kitten to look - its already in the box.