Calculating the mathematical expectation of a bet
The mathematical expectation of any bet is defined as follows: the sum of all possible gains and losses multiplied by their relative probabilities. Written as a formula, we have:
e = mathematical expectation
w = gain on the winning bet
p = probability of the win
v = value of the loss
l = probability of the loss
Let's apply that to a few of examples: a simple coin-toss game, an "unbalanced" coin-toss, the single number roulette bet mentioned on the house edge page and finally the even-money outside bets under European roulette rules.
• Coin toss 1:
The payoff for a win is even money, or 1 to 1, and the probability is 50%, or ½, for both win and loss - giving us the following formula:
The mathematical expectation is 0. Neither side has an advantage.
• Coin toss 2:
In this game, you bet one dollar, pay your opponent a dollar when you lose, but he pays you only ninety cents ($0.90) when you win. That leads to the following formula:
The mathematical expectation is - 0.05, or −5%.
• Single number roulette bets:
Winning bets are paid at 35 to 1, probability of a win is 1/37 (your number out of the total of 37 numbers) and probability of a loss therefore 36/37. That leads us to this formula:
The mathematical expectation is - 0.027, or −2.7%.
• "Even-money" European roulette bets:
Winning bets are paid at even money and the probability is 18/37. However, if the ball falls in the "zero" compartment, half the bet is returned, and the probability of zero is 1/37. This gives us the following calculation:
The mathematical expectation is - 0.0135, or −1.35%.
Armed with the above knowledge, it's a simple matter to work out your expected loss for a given amount of wagering on each of the above bets, by multiplying together your total wagered amount and the expectation for each wager in question:
Based on a total of $1000 wagering:
• Coin toss 1: 1000 × 0 = 0. Expected loss: $0.
• Coin toss 2: 1000 × −0.05 = −50. Expected loss: $50.
• Roulette inside bets: 1000 × 0.027 = −27. Expected loss: $27.
• Roulette "even money" bets: 1000 × −0.0135 = −13.5. Expected loss: $13.5.
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