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A fair coin is flipped repeatedly until it turns up tails five times. What is the expected number of heads before that happens?

Based on the link given in the comment, I have found it can be solved using the recursion $E(n)=\frac{1}{2}(E(n)+1)+\frac{1}{2}(E(n-1))$ which is equivalent to $E(n)=E(n-1)+1$. Is it correct?

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