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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


A bet's mathematical expectation is also known as its “expected value”, usually reduced to the acronym “EV”. In this context, “expected” means “average” or “mean”: although any individual bet can win or lose its EV remains constant, and the more such wagers you make the more your overall results will converge to the bet's “EV”. As such, EV is a hugely useful tool for gamblers to be aware of.

Let's apply the above formula 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.


Calculation or simulation?


The above bets are relatively simple and can be calculated precisely using the formula at the top of the page. For more complex bets it can be easier to estimate their EV using computer simulation - the computer takes the parameters of the bet and basically plays it over and over again. The longer the simulation, the more accurate the estimated EV. Simulations can run to hundreds of millions of hands and can produce numbers accurate to several decimal places.

The EV of blackjack bets is invariably estimated in this way given their immense complexity of outcomes; an example of one such simulation is in the “is blackjack streaky?” section of the blackjack page.


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