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In Feynman's 'Lectures on Physics', I read a chapter on probability which tells that P(Head) for a fair coin 'approaches' 0.5 as no. of trials that we take goes to infinity (well, I tossed the coin 50 times & got heads 17 times, instead of 25 :-) ...). Can someone elaborate?

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    The definition of a "fair coin" is that it is equally likely to fall heads and tails (and a miniscule likelihood of landing on its edge and staying there). That means, the *assumption* is that $P(Head) = 0.5$. Experimentally, the probability of landing heads is the number of successful outcomes divided by the number of experiments; so if you perform $n$ trials, and compute $h/n$ ($h$ the number of heads), you expect $h/n\to P(h)$ as $n\to\infty$. $n=25$ is very far from $\infty$, of course...2011-04-05
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    so the general assumption that P(H)=P(T)=0.5 is taken just for the sake of brevity or what?2011-04-05
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    @Amit: Again: by *definition*, a "fair coin" is one in which $P(H)=P(T)$. Assuming that $P(E)$ is negligible (landing on its edge), which is reasonable for practical purposes, this gives $P(H)=P(E)=0.5$. But probability of 1/2 does *not* mean that in any particular experiment you will *always* get half the coin tosses heads and half tails; it means that *in the long run* you expect to get as many heads as tails. That is, if a coin is "fair" (under the above definition), and you perform an experiment with $n$ tosses, you expect $h/n$ to be "close to 0.5", with "how close" proportional to $1/n$.2011-04-05
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    If you want to derive this from physical laws, as input you'll need two main ingredients: that the coin is symmetrical (of course this would give you the problem that you couldn't determine heads from tails, but ignore this!) and that there's no probability the coin could land in any configuration other than heads or tails -- say the "edge" of the coin is tapered to make standing on edge an unstable configuration. Then you compute the probability of landing in either configuration as the relative volume of the attractive basins (in state-space) for the two final configurations.2011-04-05
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    Get it more clearly now. After all we're talking about 'Probability' (& NOT 'Surety'). Hence, not getting 25 heads in my experiment of 50 tosses was not at all wrong result (to be lost in) or something. Thank you again, sir. And, +1 for "**how close is proportional to 1/n**"2011-04-05
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    Arturo, isn't "how close" supposed to be proportional to $1/\sqrt{n}$?2011-04-05
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    @Michael Lugo: I'm sure you can make it precise; I just meant "the bigger the $n$, the closer you get", and was running out of space in the comment. Sorry if that meant I said something false.2011-04-05

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