Mathematical expectation



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= (w × p) + (-v × 1)


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, the even-money outside bets under European roulette rules and the "perfect pairs" blackjack side bet.


Coin toss 1


The payoff for a win is even money, or 1 to 1, and the probability is 50%, or 1/2, for both win and loss - giving us the following formula:


(1 × ½) + (-1 × ½) = 0


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:


(0.9 × ½) + (-1 × ½) = - 0.05


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:


(35 × (1/37)) + (-1 × (36/37)) = - 0.027


The mathematical expectation is - 0.027, or 2.7%.



Here are two wagers with multi-tiered outcomes:


"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:


(1 × (18/37)) + (-1 × (18/37)) + (-½ × (1/37)) = - 0.0135


The mathematical expectation is - 0.0135, or 1.35%.


"Perfect pairs" blackjack sidebet


In this bet, the payoff is received if a pair is dealt. The size of the win depends on the kind of pair card received - mixed colour (AH + AC/AS), matching colours (10H + 10D) or "perfect" (6C + 6C). The payoffs and their probabilities are listed in the table below, based on a six-deck game:


HandPayoffProbability
Mixed pair5 to 112 / 311
Coloured pair10 to 16 / 311
Perfect pair30 to 15 / 311
No pair-1288 / 311



This leads to the following calculation:


(5 × (12/311)) + (10 × (6/311)) + (30 × (5/311)) + (-1 × (288/311)) = - 0.05787



The expectation is -0.05787, or a house edge of 5.5787%.


As you can see, the method of calculation for the more complex bets is the same as the simple ones.


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.

Blackjack "perfect pairs" side bet: 1000 × 0.05787. Expected loss: $57.87.


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