The intuition at $p = \frac{1}{2}$ is based on symmetry.
If Calvin wins a game with probability $p = \frac{1}{2}$, then Hobbes also wins a game with *the same probability* $q = 1 - p = \frac{1}{2}$. Then the respective probabilities $P(C)$ and $P(H)$ of each player winning the whole match *must also be same*: $P(C) = P(H)$.
Then the particular value $P(C) = P(H) = \frac{1}{2}$ follows from the assumption that either Calvin wins and Hobbes does not, or else Hobbes wins and Calvin does not (the events are *mutually exclusive* and *exhaustive*), so that $P(C) + P(H) = 1$.
viäränlaenen
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2022-05-09T13:01:11Z (almost 2 years ago)