Expected Value

Expected value (or expectation value or EV) is a term used in probability theory and statistics and is defined as the sum of all possible values for a random variable, each value multiplied by its' probability of occurrence. In terms of betting, EV is what you can expect to win or lose if you were to bet on the same outcome many times with the same stake.

EV is calculated as follows,

EV = [(probability of winning) X (amount won per bet)] + [(probability of losing) X (amount lost per bet)]

Using a coin as an example. Assuming that the coin is fair, when the coin is tossed and it lands on heads you lose £1 but if it lands on tails you win £2. On average you would expect to win once and lose once every two tosses. However, when you win you get £2 but when you lose you lose £1, therefore, in the long run you will gain £2 + (-£1) = £1 for every two tosses so on average you will win half that, 50p, every time the coin is tossed.

You can then use this information to estimate how much you will win over 1000 tosses. At, on average, 50p per toss you expect to win 50p X 1000 = £500 over 1000 tosses.

To calculate expected value, add up the sum of the probabilities of each event multiplied by the payout for each event. Using the above example, the probability of either heads or tails is 1 / 2 for each outcome so the expected value is,

EV = [(1/2) X (£2)]+[(1/2) X (-£1)]
= [£1] + [-50p]
= 50p

Using roulette as another example. Assuming you are playing on a European roulette table where there are 37 numbers (only 0 and not 00 as on an American roulette table) a bet on a winning number pays out 35/1 in which your original stake is not lost and 35 times the stake money is won. Considering all 37 possible, the expected value of the profit from resulting from a £1 bet on a single number is the sum of the odds of winning times what you will win multiplied by the odds of losing times what you will lose.

Probability of winning = 1 / 37
Amount won per bet = £35
Probability of losing = 36 / 37
Amount lost per bet = -£1

EV(winnings per bet) = [(1/37) X £35] + [(36/37) X -£1] = -0.027

Therefore, you can expect to lose, on average, about 3p for every £1 you bet at the roulette wheel.

Conclusion, you should not expect to make a long term profit from bets with a negative expected value. Look for opportunities where there is a positive expected value.