Smart Bettors Understand Expectation Value (EV)

If you have done any reading about gambling theory in the past, you likely came across the term expectation value, commonly abbreviated EV. It is a fundamental concept that should be understood if you're going to be serious about your betting.

In words the concept is simple: it is the amount you would expect to win or lose on one instance of a bet, averaged over an infinite number of instances of that bet. To determine what the exact expectation value of a bet is requires exact knowledge of

the odds of winning and
the amount you will be paid off for winning.

The simplest example is a bet that has a 50% chance of winning - such as a coin flip - with 2-1 odds for winning. You can flip a coin as many times as you want, and you must bet $1 for each flip. If it lands heads you are paid $2. If it lands tails you lose your dollar. Even without math you can see that this would be a good deal, but let's do the math to see just how good.

First let's list all of the possibilities and their outcomes, then add them up

50% of the time it will be heads for $2
50% of the time it will be tails for (-$1)

Mathematically

(.50)($2) + (.50)(-$1) =
($1) + (-$0.50) =
$0.50

Thus our expectation value for each coin flip is $0.50. You will not win $0.50 on any single coin flip, you will win $2 or lose $1, but the expectation value of the bet is $0.50. If we flip the coin 1,000 times, we would expect to win (1,000)($0.50) = $500.

Another Expectation Value Example

Let's look at another, slightly more complex, scenario: An American Roulette wheel.

American roulette has 38 numbers on the wheel. 18 are red, 18 are black, and 2 are green. A bet on red (or black) pays off at even money. If you bet $1 on red and win, you will be paid $1.

What, then, is the expectation value of of betting $1 on red?

As there are 18 red numbers that win and 20 non-red (black and green) numbers that lose, we can represent the equation thus:

(18/38)($1) + (20/38)(-$1) =
(.4737)($1) + (.5263)(-$1) =
$0.4737 + (-$.5263) =
(-$0.0526)

You will expect to lose, on average, a little over 5 cents for every $1 you bet on red at an American Roulette table. Of course, you would not lose 5 cents on any single bet. You will win $1 or lose $1. But if you made 1,000 bets on red at $1 you would expect to lose about $53.

If you base all of your decisions purely on math and logic, you should never make a bet that has a negative expectation value (such as our roulette example). However, there may be other reasons to make such a bet. Perhaps you find playing roulette entertaining. Perhaps there's a gorgeous redhead at the roulette table and you're trying to get her attention. These can have a positive life quality expectation, even if the bet doesn't have a positive expectation value. But if your goal is to make money, you should stay away from bets with a negative expectation value (-EV)