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Is it possible to determine that a coin is biased or not, by tossing it a number of times ?

I am sure that this problem has been studied,I am interested to know about the mathematics behind this problem.

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You may want to look at –  joriki Oct 13 '11 at 7:24
This comic strip makes fun of the fact: –  André Caldas Oct 13 '11 at 12:19

2 Answers 2

up vote 5 down vote accepted

No "determined" result can be got this way, you get at most is a "likelihood" ... Next question is then, whether you feel that that likelihood is "sufficiently" high for your needs in the real/physical/social world...

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I'll give a concrete answer here, since neither of the previous answers is exactly concrete.

For a fair coin, after $N$ flips, you expect to see $N/2$ heads. How far do you expect to get from $N/2$ heads? Well, again for a fair coin, if you repeat this expermient multiple times, you expect to be within $n = \sqrt{N}/2$ of this expected value about 2/3 of the time. You expect to be within $2n$ about 95% of the time, and within $3n$ about 99% of the time. If you are outside one of these intervals, you start thinking the coin might be unfair (and there are methods, as discussed above, to quantify the probability of fair vs. degree of unfairness). These are called "Confidence Levels".

The tricky bit is, if the coin is only a little unfair (e.g. on average you get 501 heads out of 1000 coin flips), you have to make a very large number of coin flips to definitively distinguish this case from the fair case.

Technically, you have a probability distribution $P(p)$ which tells you the probability that the coin is biased p:(1-p) heads-to-tails given your state of knowledge. As the number of coin flips goes up, that probability distribution gets narrower (and is always centered around {# of heads}/{# of flips}).

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