Expected Flips for Consecutive Heads
You have a fair coin. Let $X_i$ be the number of flips required to see $i$ consecutive heads for the first time.
Compute $\mathbb{E}[X_1]$, $\mathbb{E}[X_2]$, and $\mathbb{E}[X_3]$.
For each, set up a recursive equation and solve it. Can you spot the pattern and conjecture a formula for general $\mathbb{E}[X_k]$?
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