On Triumvirate

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When last they met, the Baron invited Sir R----- to join him in a wager involving a sequence of coin tosses. At a cost of seven coins Sir R----- would receive one coin for every toss of the coin until a run of three heads or of three tails brought the game to its conclusion.

To evaluate its worth to Sir R----- we begin with his expected winnings after a single toss of the coin.
\[ \begin{align*} \mathrm{E}_T[n] &= \tfrac12 \times \left(1+\mathrm{E}_{TT}[n]\right) + \tfrac12 \times \left(1+\mathrm{E}_{H}[n]\right)\\ \mathrm{E}_H[n] &= \tfrac12 \times \left(1+\mathrm{E}_{T}[n]\right) + \tfrac12 \times \left(1+\mathrm{E}_{HH}[n]\right) \end{align*} \]
Note that we're exploiting the fact that if the next coin shows a different face to the most recent one then the current coins cannot be part of a run of three and Sir R-----'s further expected takings are identical those that he had at the outset. We can similarly calculate the expected prize after two tosses of the coin with
\[ \begin{align*} \mathrm{E}_{TT}[n] &= \tfrac12 \times 1 + \tfrac12 \times \left(1+\mathrm{E}_H[n]\right)\\ \mathrm{E}_{HH}[n] &= \tfrac12 \times \left(1+\mathrm{E}_T[n]\right) + \tfrac12 \times 1 \end{align*} \]
Finally, at the start of the game we have
\[ \mathrm{E}[n] = \tfrac12 \times \left(1+\mathrm{E}_T[n]\right) + \tfrac12 \times \left(1+\mathrm{E}_H[n]\right)\\ \]
Noting that the probabilities of tossing heads and tails are equal, we can simplify these formulae by exploiting the identities
\[ \begin{align*} \mathrm{E}_H[n] &= \mathrm{E}_T[n]\\ \mathrm{E}_{HH}[n] &= \mathrm{E}_{TT}[n] \end{align*} \]
yielding by recursion
\[ \begin{align*} \mathrm{E}_{TT}[n] &= \tfrac12 \times 1 + \tfrac12 \times \left(1+\mathrm{E}_T[n]\right)\\ &= 1 + \tfrac12 \times \mathrm{E}_T[n]\\ \mathrm{E}_T[n] &= \tfrac12 \times \left(1+\mathrm{E}_{TT}[n]\right) + \tfrac12 \times \left(1+\mathrm{E}_{T}[n]\right)\\ &= 1 + \tfrac12 \times \left(\mathrm{E}_{TT}[n] + \mathrm{E}_{T}[n]\right)\\ &= 1 + \tfrac12 \times \left(\left(1 + \tfrac12 \times \mathrm{E}_T[n]\right) + \mathrm{E}_T[n]\right)\\ &= \tfrac32 + \tfrac34 \times \mathrm{E}_T[n] \end{align*} \]
and consequently
\[ \begin{align*} \tfrac14 \times \mathrm{E}_T[n] &= \tfrac32\\ \mathrm{E}_T[n] &= 6 \end{align*} \]
The expected number of tosses, and Sir R-----'s bounty, is therefore
\[ \mathrm{E}[n] = 1 + \mathrm{E}_T[n] = 7 \]
meaning that the game is perfectly fair and I should have suggested to Sir R----- that he could take it or leave it!
\(\Box\)

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