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Posted by Travis on 2006-09-26 16:10:22 +0000

the sample number is not relavent - you are performing an action which has 2 possible outcomes (unless you live on the discworld and coins land on there edge) - you could flip a coin a million times and it could come up heads every time, the probability of it coming up heads is still 1/2 - the coin has no way of knowing that it has just landed heads a million times. Its highly improbable that it would land heads a million times in a row but that does not change the probability of heads or tails which remains 1/2

Posted by Miriam on 2006-09-26 16:13:29 +0000

Did you watch the thing on the Discovery Channel on Sunday afternoon, too?

Posted by Travis on 2006-09-26 16:17:00 +0000

nope - just like gambling

Posted by pchippy on 2006-09-26 16:33:09 +0000

It's really more of a statistical question than a probabilistic one. In essence, NO number of sampled coin tosses will be enough to measure precisely the probability of landing heads for a given toss. Ten tosses would give you some sense of whether a coin is a fair coin--but not a very good sense. A hundred tosses would be better. A thousand tosses would be better still. But you'll never get an absolutely certain answer. You might toss a coin a hundred times in a row and get heads every time, but it might still be a perfectly fair coin. But the larger the sample size, the less likely it is that your experimental results will be way off the correct figure. You can use Pascal's triangle to calculate the possible results for N tosses of a fair coin. For ten tosses there will be 2^10 possible outcomes, or 1024 if I'm doing my math correctly. One would expect to toss exactly five heads out of ten tosses in 252 out of 1024 trials, or almost a qarter of the time. One would get between four and six heads in 672 out of 1024 trials, or about two thirds of the time. One would get between three and seven heads in 912 out of 1024 trials, or almost three-quarters of the time. The odds that the fair coin will land two to eight heads in a ten-flip trial are 1002 out of 1024, or about 98%. The odds that the coin will land heads one to nine times in a ten-flip trial are 1022 out of 1024, or better than 99.8%. BUT even the absolutely perfect and fair coin has a 1 in 512 chance of landing ten heads or ten tails in a row. Not exactly the information you were looking for, right? Sorry. I'm not a statistician myself. Did the original question have anything to do with football?

Posted by tgl on 2006-09-26 17:11:06 +0000

You don't determine the probability of heads or tails through empirical means. I think Travis is onto the answer. You calculate the probability beforehand, knowing that there are two possible outcomes, heads or tails. So it's 50/50 or 1/2 or however you want to described it. So, 1 coin flip is 1 too many to determine the odds. It's "enough" to calculate the probability using combinatorics. Same for something more interesting: you determine the probability of 5 head and 5 tails on 10 flips without actually flipping a coin once. Or the probability of 10 heads and no tails. The empirical evidence only confirms your theoretical answer, it does not generate it. See here: fourmilab.ch Probability of 5 heads and 5 tails on 10 flips: 0.0198 Probability of 10 heads and 0 tails on 10 flips: 2.75e-7

Posted by tgl on 2006-09-26 17:13:42 +0000

Now that I re-read BQ3K's post, he's wondering how much is "enough" empirical evidence to prove that we calculated 50/50 properly... Maybe pchippy's answer is satisfactory.

Posted by tendiamonds on 2006-09-26 17:20:50 +0000

I would paraphrase, but Wikipedia does a nice job here.

Posted by tommy on 2006-09-26 18:05:29 +0000

I'll take a stab at a quick outline (see the Wikipedia article for more details)... Rephrased question: "I have a coin... how many times to I have to flip it to prove it's not biased towards heads or tails?" PChippy was right, no amount of flips is enough to prove beyond any doubt that the coin is unbiased. But, you can say how confident you are that the coin is unbiased to within some tolerance. The more flips, the higher your confidence, and/or the better the tolerance. With only a handful of flips, you can maybe say "There's at least a 10% chance that this coin is no more than 15% biased" With a lot of flips, you can say something like "I'm 99% sure that this coin is no more than 1% biased". This is a similar problem to a political poll, which is (or at least should be) always accompanied by something like "We're 95% confident that these numbers are no more than 5% off".

Posted by bizquig3000 on 2006-09-26 18:10:24 +0000

Liz got me some silly pop culture book about mathematics. There was one chapter about Kenneth Arrow and how there's no mathematical thing as a fair election process. It also had some silly comments about maths used in "fairness" (the development of fairness among children being the subject of Liz's PhD thesis). Whilst talking about this chapter, we ended up bullshitting about probability shortly thereafter. I couldn't/wouldn't have thought of "margin of error" 10D. Thanks.

Posted by tommy on 2006-09-26 18:13:27 +0000

OMG! Forgot to add this. Rule of thumb is if you want a precision of P with about 95% confidence, then flip the coin 1/(P^2) times. So, to be 95% sure your numbers are less than 10% off, flip the coin 100 times, because 1/(0.1^2) = 1/0.01 = 100. For less than 5% off, flip it 400 times. For less than 1% off, flip it 10,000 times. And so on...

Posted by ConorClockwise on 2006-09-26 18:27:29 +0000

Right on, tommy. Forgot that too. Suddenly my mind is lost, wondering about standard deviations... doing some quick calcs