In the previous problems, if we let \( X \) denote the value of Scrooge’s winnings, and let \( X_n \) denote the value of Scrooge’s winnings during each round, we have
\[ X = X_1 + X_2 + X_3 + X_4 + \ldots \]
Even though there are infinitely many terms (countable), only finitely many of them are non-zero. Hence, this sum makes sense under every scenario. In each game, \( E[X_i] = 0 \), since it is a fair value.
The linearity of expectation tells us that if \( X = X_1 + X_2 \), then we have \( E[X] = E[X_1] + E[X_2] \). As such, we would be very tempted to claim that
\[ E[X] = E[X_1] + E[X_2] + E[X_3] + \ldots = 0 \]
However, as we have seen, that is not the case. Only in the second problem, did we have \( E[X] = 0 \).
What’s the reason for this?