Some coin flip/Bayes theorem questions

[magic_man]

I always knew about Bayes theorem, but had never really seen it in use until a recent post on coin flips. These questions got me thinking, and I've come up with the following situations/questions. Can anyone help out?

1) You flip a coin 10 times, and the result is tails 8/10 times. Is it possible to calculate the odds that you somehow have an "unfair" coin, without knowing in what way it is unfair? In other words, what are the odds that you DON'T have a "fair" coin?

2) Same situation as above, but a friend who works at the mint has also told you that about 1/1000 coins is made unbalanced, so that it is unfair. However, it's possible that these errors are biased one way or another, but you're not sure which. In other words, you can't just assume that 1/2000 coins is biased towards heads, and 1/2000 is biasad towards tails.


My point in the two situations above is that you have additional information, but it is not complete. I.E. in scenario 1, you do not know to what extent the coin is unfair, and in scenario 2 you know neither the extent nor the direction, but you do know the odds that unfairness occurs.

Is there any way to make sense of these situations? I'm getting very curious about all this.

~Magic_Man

[gaming_mouse]

[ QUOTE ]
1) You flip a coin 10 times, and the result is tails 8/10 times. Is it possible to calculate the odds that you somehow have an "unfair" coin, without knowing in what way it is unfair? In other words, what are the odds that you DON'T have a "fair" coin?

[/ QUOTE ]

You can do a statistical test called a p-test. Essentially what you do is assume that the coin is fair, and then ask yourself how unlikely the observed results are given this assumption. If they are very unlikely, then you deduce that your assumption was incorrect, and that the coin is actually unfair. It's called a "p" test because "p" is the probability of the observed results given your hypothesis (in this case, that the coin is fair). In the scientific community, if "p" is less than .05, your results are considered "statistically significant." In other words, you have decent evidence against your hypothesis.

Now in your case, we first note that you would have been equally surprised if there had been 8 heads rather than 8 tails, and even more so if there had been 9 or 10 of either. So we need to calculate the chance that of 0,1,2,8,9, or 10 tails. This is:

2*(1/2)^10 + 2*10*(1/2)^10 + 2*45*(1/2)^10 = .1008

Thus, even with a fair coin, you would see results as extreme as these 10% of the time. This is not enough evidence to conclude that the coin is unfair.

gm