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Expected Coin Flips for Patterns

Hard · Conditional Expected Value · Quant Trader interview question · probability, expected-value, mental-math, conditional-expectation

You are trading a contract whose payoff depends on patterns in repeated flips of a fair coin, where heads and tails each have probability 50%. Consider two separate experiments, each starting from a fresh sequence of coin flips: 1. In the first experiment, keep flipping until HH appears for the first time. Let its expected number of flips be $E_{HH}$. 2. In the second experiment, start over with a fresh sequence and keep flipping until HT appears for the first time. Let its expected number of fl