# Expected Number of Runs in Coin Tosses A fair coin is tossed 21 times. A run is a maximal consecutive block of identical outcomes. For example, `HHHTTH` contains three runs: `HHH`, `TT`, and `H`. Find the expected number of runs. ### Constraints & Assumptions - Tosses are independent and each side has probability one half. - Every nonempty sequence begins with exactly one run. ### Clarifying Questions to Ask - Is the coin fair and are tosses independent? - Is a run counted once regardless of its length? ```hint Count changes between neighbors After the first toss, a new run begins exactly when the next toss differs from the previous one. ``` ### What a Strong Answer Covers - An indicator-variable representation of the run count. - The probability of a change at each adjacent pair. - Use of linearity of expectation without assuming the indicators are independent. - The final numerical expectation. ### Follow-up Questions - What is the expectation for `n` tosses of a coin with head probability `p`? - How would you compute the variance of the number of runs?