Suppose flipping a coin with probability $p$ to get a head. Suppose we flip it until a head appear. What is the mean number of flip required getting a head? (Better to use conditional expectation to show the mean)
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If $E$ is the expected number of flips, then we have following relation $$ E = p\cdot 1+ (1-p)\cdot(1+E)$$ because with probability $p$ we succeed at first try and with probability $1-p$ we have "wasted" one try and start again. Once we agree that $E$ is finite, this produces $$ E = \frac1p.$$ |
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