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If I have a coin then chances of getting a $head$ or a $tail$ is $50-50$. But why don't we take in account the case where coin is neither head and tail, where coin is standing upright, or it is making an angle with ground? Some people may object that when we are talking about probability of getting a $head$ or $tail$, what we mean is that if we are doing experiment in a controlled room, where there is no matter except the coin and that coin is fully uniform, it is $50-50$ chance that I will get a $head$ or $tail$. This argument is not a very satisfying one to me, it don't even answer the question:How can we know that only two possible cases are there? Why consider only head and tail? Actually it raises one more question: Why the probability of getting a $head$ or $tail$ is $50-50$?

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    $\begingroup$ When we speak of things like fair coins in math we are talking about idealized situations, where we can declare the probability as we see fit. Real world situations may be modeled by such things but it's rare that the modeling is perfect. $\endgroup$ – lulu Jan 12 '20 at 12:07
  • $\begingroup$ Moreover, it will occur extremely rare that the coin lands upright. If it actually occurs , we just do not count the flip. $\endgroup$ – Peter Jan 12 '20 at 12:08
  • $\begingroup$ A real coin barely will give the exact probability, only a more or less good approximation. Usually, deviations are so small that we cannot observe it in practice. $\endgroup$ – Peter Jan 12 '20 at 12:10
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Usually, when mathematicians talk about 'a coin flip' they mean the idea of an ideal coin flip, where the probability of getting 'head' or 'tails' is exactly one half for either of the possibilities. This may or may not be a good model for the outcomes of a real coin flip. But there is nothing stopping you from thinking of a model where the outcome can either be 'head', 'tails' or 'indeterminate', where also 'head' or 'tails' may not have the same probability.

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In real life, as opposed to in a problem in an elementary textbook on probability, a coin is not a perfect generator of $50$-$50$ odds.

"Probability of a tossed coin landing on edge," Daniel B. Murray and Scott W. Teare, Phys. Rev. E 48 (Oct. 1993) gives a chance of approximately $1/6000$ that if a U.S. nickel (five-cent coin) is tossed, it will land on edge.

"Dynamical Bias in the Coin Toss," Persi Diaconis, Susan Holmes and Richard Montgomery, SIAM Review Vol. 49, No. 2 (Jun., 2007), pp. 211-235 concluded that with $0.51$ probability, a coin lands with the same side facing up as before it was tossed.

"How random is the toss of a coin?" Matthew P.A. Clark, MBBS and Brian D. Westerberg, MD, CMAJ. 2009 Dec 8; 181(12): E306–E308 found that some people are able to manipulate the toss of a coin.

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