\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
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 (
    ! e \(\pi\) ( ) % AC sin cos tan 7 8 9 / \(\sin^{-1}\) \(\cos^{-1}\) \(\tan^{-1}\) 4 5 6 * \(x^2\) \(\sqrt{\phantom{2}}\) xy 1 2 3 - ln exp log 0 . = +
    ).
  • Interpret the result in terms of the player’s average outcome per game.

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