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In a game of chance, a player bets \(\dollar 1\) on a single number in a classical roulette wheel numbered from 0 to 36. If the chosen number comes up, the player wins 35 times their bet plus their bet back, receiving a total payout of \(\dollar 36\); otherwise, they lose their bet. In gambling, the gain is defined as the payout minus the cost to play.
  • Calculate the expected gain \(E(X)\) of the player.
  • Interpret the result in terms of the player’s average outcome per game.
\FieldText{1}{
  • Let \(X\) represent the gain of the game. The probability distribution of \(X\) is
    \(x\) -1 35
    \(P(X=x)\) \(\frac{36}{37}\) \(\frac{1}{37}\)
    .
    \(E(X) = (-1) \cdot \frac{36}{37} + 35 \cdot \frac{1}{37} = -\frac{36}{37} + \frac{35}{37} = -\frac{1}{37}\)
  • Since \(E(X) = -\frac{1}{37}\), we expect the player to lose an average of approximately \(\dollar 0.027\) per game.
}