Compare and contrast the following two betting schemes for roulette, which are sometimes used by gamblers.

Double on each loss Bet \(\$ 1\) on black. If black comes up, you win \(\$ 1\), and the game is over. If black does not come up, double your bet (\$2). If black now comes up, your net winnings are \(\$ 1\) (lose \(\$ 1\), then win \(\$ 2\) ), and the game is over. If black does not come up, double your bet (\$4). If black now comes up, your net winnings are \(\$ 1\) (lose \(\$ 1\), lose \(\$ 2\), then win \(\$ 4)\). Continue with the doubling procedure until you eventually win; in every case, your net winnings are \(\$ 1\).

Double on each win Bet \(\$ 1\) on black. If black does not come up, you lose \(\$ 1\), and the game is over. If black comes up, double your bet (\$2), and play again. If black does not come up on the next spin, you lose \(\$ 1\) (win \(\$ 1\), then lose \(\$ 2\) ), and the game is over. If black comes up, double your bet (\$4) and play again. If black does not come up on the next spin, you lose \(\$ 1\) \((+1+2-4)\), and the game is over. If black comes up, double your bet ( \(\$ 8\) ), and play again. If black does not come up on the next spin, you lose \(\$ 1(+1+2+4-8)\), and the game is over. If black does come up, double your bet \((\$ 16)\) one last time. If black does not come up on the next spin, you lose

\(\$ 1(+1+2+4+8-16)\), and the game is over. If black does come up, the game is over and you win \(\$ 31(+1+2+4+8\) \(+16)\). Using this scheme, you are risking a maximum loss of \(\$ 1\) for a chance at winning \(\$ 31\).