You're at the table. Red or black. Odd or even. 1-18 or 19-36. These even-money bets feel like a coin flip. You're wrong.
European roulette: 18 red, 18 black, 1 green zero. 37 total. Your bet on red wins if the wheel lands on red (18 times out of 37). Your probability of winning is 18/37 = 48.65%. Your probability of losing is 19/37 = 51.35%.
If you bet 100 EUR and win, you get 200 EUR (your bet back plus 100 EUR winnings). If you lose, you get 0. Your expected value is (0.4865 x 200) + (0.5135 x 0) minus 100 (your wager) = minus 2.70 EUR. House edge is 2.70%.
American roulette: 18 red, 18 black, 1 zero, 1 double-zero. 38 total. Your probability of winning is 18/38 = 47.37%. Your probability of losing is 20/38 = 52.63%.
Expected value is (0.4737 x 200) + (0.5263 x 0) minus 100 (your wager) = minus 5.26 EUR. House edge is 5.26%.
The Single Number Trap
If you bet on a single number (17, say), European pays 35:1 (you win 3,500 if you bet 100 and hit). American also pays 35:1. But your odds are different.
European: 1/37 = 2.70% chance. Payout at true odds would be 37:1 (you'd get your 100 back plus 3,700 winnings). You get 3,500, so the house keeps 1.08%.
Actually, you lose money on the true calculation because of the zeros. On European, a single-number bet has a house edge of 2.70% (the zero disadvantage is spread across all bets equally). On American, with two zeros, the house edge is 5.26%.
The Side Bet Trap
American roulette often has a special bet: the five-number bet (0, 00, 1, 2, 3). This bet pays 6:1. The odds of hitting one of those five numbers out of 38 is 5/38 = 13.16%. True odds would be 38/5 = 7.6:1. You're getting 6:1, so the house has an advantage.
Calculation: (5/38 x 6) minus 1 (your bet) = negative 0.2105. House edge is 21%.
Wait, 21%? No, I made a unit error. Let me recalculate. If you bet 100 and win, you get 600. If you lose, you get 0. Expected value = (0.1316 x 600) + (0.8684 x 0) minus 100 = minus 21.05 USD. House edge on that specific bet is 21.05%.
No casino offers a five-number bet on European roulette. Only American. It's predatory.
The Geography
European roulette dominates in Europe, Asia, Australia. American roulette dominates in the U.S. and Las Vegas (for historical reasons). Monaco has only European wheels. Atlantic City has American wheels (though some casinos offer European as a premium game).
If you find yourself in a casino with a choice, always choose European. The 2.7% edge is still bad, but it's half the American edge.
The Math Is Unforgiving
A player betting 100 USD per spin on European roulette for 1 hour (roughly 60 spins) expects to lose approximately 160 USD. On American, approximately 315 USD. Same time, same activity, vastly different cost.
This is not entertainment value. This is mathematics.
