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As per basic probability theorem getting equal number of heads and tails when coin is tossed $10000$ times is = $\frac{\binom{n}{r}}{2^n} = \frac{\binom{10000}{5000}}{2^{10000}}$

This value is not equal to or close to $0.5$

But if one does the same experiment simulation, then probability will be close to $0.5$ (with $10000$ tosses)

Can you explain why is there a difference in probability ?

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    $\begingroup$ Have you done the simulation? $\endgroup$
    – user619894
    Commented May 25, 2020 at 11:50
  • $\begingroup$ No, I followed this link for simulation data: link $\endgroup$
    – inode
    Commented May 25, 2020 at 14:29
  • $\begingroup$ I think you misunderstood the results, nowhere do they give the probability of seeing exactly the same amount of heads and tails. $\endgroup$
    – user619894
    Commented May 25, 2020 at 15:49

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The probability being close to $1/2,$ has to do with relative error; namely since $$ \frac{\binom{n}{n/2}}{2^n}\approx \frac{1}{\sqrt{\pi n}} $$ by Stirling approximation, the probability of being within $O(\sqrt{n})$ terms, i.e. within $O(\frac{\sqrt{n}}{2^n})$ of the of the exact $1/2$ probability value goes to $1$ as $n$ increases.

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