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As You have one more coin than me, if you don't throw more heads than me, then you must throw more tails than me.

This means that the probability of you throwing more heads then me plus the probability of you throwing more tails then me is equal to one.

By symmetry, the probabilities for heads and tails must be equal and so the probability that you throw more heads is \(\frac{1}{2}\)

Extension

You toss \(n\) fair coins, and I toss \(m\) fair coins. What is the probability that you get more heads than I do?