Hello,

Let X be the number of heads that you get. This follows a binomial distribution of parameters (n,p), where p=0.5, since it's a balanced coin.

The proportion H of heads is given by

You want to prove that

The mean of a binomial distribution is np.

So

So

But we know by Chebyshev's inequality :

The variance of a binomial distribution is .

So

Hence

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