If your friends are gamblers you can be pretty sure of coming across the term “expected value” at some point. If you hang around with poker players in particular you’ll come across it almost immediately. In the latter case, where internet forums have bled their influence into the cloth of the language, it will often be abbreviated to EV.

Mathematicians sometimes refer to it as simply “the expectation”, or as the “mean value” or “first moment”.

If you wondered what exactly that is, then wonder no longer: the expected value of an event is defined as the probability-weighted average of all the possible outcomes of that event.

Here “event” is going to refer to anything that could happen in which there are multiple possible outcomes with known probabilities, a definition that covers pretty much every bet in the casino. So, the general equation for this for a range of outcomes X1 to Xn with is basically:

Expected Value = (Probability of X1 x Value of X1) + … + (Probability of Xn x Value of Xn)

If that doesn’t click immediately in your brain, don’t worry. In practice the maths is simple addition, fractions and multiplication. And if you like a punt, that kind of maths is important to know.

A more simple way to think about EV more intuitively is as the average outcome of running the same event a functionally infinite number of times. For example, you rolled an unbiased dice over and over and got \$1 for each pip showing on the dice (\$1 for a roll of one, \$2 for a roll of two, etc…) then you could do the maths above or you could imagine rolling the dice over and over and tracking the results.

Since the sides all come up an equal number of times in the long run, as the number of rolls gets bigger and bigger each side of the dice will come up ⅙ of the time. As a result, your average win will get closer and closer to \$3.5 per roll. This is the expected value of this bet.

The reason for this is that the average roll of a dice is equal to the average of the six possible rolls – or to use the formula in the section above:

EV = (\$1 x ⅙) + (\$2 x ⅙) + (\$3 x ⅙) + (\$4 x ⅙) + (\$5 x ⅙) + (\$6 x ⅙) = \$3.5 per roll on average.

Or, more simply put: in the long run your profits will tend towards \$3.5 per roll.

### A Worked Example

Here’s a more likely example to clarify. You’re playing roulette at your local casino and want to place \$10 on red and \$10 on Red 3 on a wheel with only a single zero. Here’s how you work out the expected value of the upcoming spin of the wheel.

First gather all your information, your outcomes and their odds. A roulette wheel with one zero on it gives odds of 1 in 37, so odds of 1/37 for each possible slot on the wheel.

The possible outcomes are: we win both bets, we win the bet on red but lose the bet on 3, or we lose both bets.

The odds for these are as follows:

• We win both bets only if the ball lands on Red 3. That is 1 outcome in 37 equally likely outcomes so 1/37. If this happens we double our Red bet \$10 profit and also get a 35-1 payout on the Red 3 bet for a \$350 profit making and outcome of \$360 profit total.
• We win the red bet if the ball lands on any other red. That is 17 outcomes in 37 – the eighteen red numbers minus the Red 3. If this happens we to break even (lose the \$10 on Red 3 win \$10 on Red). So probability of 17/37 and an outcome of \$0.
• We lose both bets: 19 outcomes in 37 (all the black numbers and the zero) in this scenario we lose \$20 with a probability of 19/37.

So the EV calculation for this spin looks like this (with outcomes rounded to two decimal places):

EV = (1/37 x \$360) + (17/37 x \$0) + (19/37 x –\$20)
EV = \$9.72 + \$0 + –\$10.27
EV = –\$0.55

In other words if you were to make this bet over and over again, your losses would tend towards \$0.55 per roll. You can think of that as a house edge of 2.75% (i.e. \$0.55 / \$20), or you can imagine it as expecting to lose \$5,500 to the casino if you make this bet 10,000 times.

The thing to note is that in the short run, your actual results are likely to be different from your expected value (how different those results are likely to be depends on the “variance” of the game).

In the shortest run of one spin for example, you will end up with either a \$20 loss, your money back, or an extra \$360 in profit. But in the long run that will tend towards –\$0.55 per spin.

Mind you, to paraphrase Keynes, in the long run we’ll all be dead.

To find out more about roulette, how to play the game and figure out how to make the best of EV why not check out our comprehensive roulette guide here.

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