Expected Value for the Casino Gambler

The importance of expected value in gambling

It’s probably obvious to most poker players or sports bettors who are serious about their gambling why “expected value” is important. But expected value is equally important for the casino gambler, and this holds true even if the gambler in question is just a recreational gambler. You might have heard the phrases “house edge” and/or “payback percentage” when talking about casino games. These are just functions of expected value, which is a way of looking at the value of a bet.

  • If the net expected value of a bet would result in a loss, the expected value is negative.
  • If it would result in a win, the expected value is positive.

Here’s how expected value works and why it’s important for the casino gambler:

What Is Expected Value and How Does Probability Play into It?

Probability is one of the factors that go into determining the expected value of a bet. The other factors are how much you’re risking versus how much you stand to win. Probability is just a way to measure the likelihood that something will occur. You should understand a few facts about probability:

  • An event’s probability is always a number between 0 and 1.
  • An event with a probability of 0 will never happen.
  • An event with a probability of 1 will always happen.
  • Probability can be expressed as a fraction, a decimal, a percentage, or as odds.

The simplest example of a probability that I can think of is flipping a coin. Probability is the number of ways something can happen divided by the total number of possible outcomes. With a coin toss, you have 2 potential outcomes total – heads or tails.

    • If you want to know the probability of getting heads, you look at the number of possible outcomes that equal tails.
  • On a 2-side coin, there’s only one outcome that’s tails.
  • The probability of getting tails when you flip a coin, therefore, is 1/2. You can express this as a fraction, as I just did, or as a decimal – 0.50.
    • You can also express this as a percentage – 50%.
    • Or you can express it as odds  — 1 to 1. (With odds, you COMPARE the number of ways something can’t happen with the number of ways it can). With the online roulette wheel, you have 38 possible outcomes.

Here’s Something Else to Understand about Probability

If you want to calculate the probability of multiple things happening, you either multiply or add the probabilities together, based on whether you want to know if A AND B will happen or you want to know if A OR B will happen.

For example, if you flip a coin twice, and you want to know the probability that you’ll get tails twice in a row, you multiply the probability. (That’s because you want to know the probability that you’ll get tails on the first toss AND get tails on the 2nd toss.)

That’s 1/2 X 1/2, or 1/4.

And again, that can be expressed as 1/4, 0.25, 25%, or 3 to 1.

Suppose, though, you want to know the probability that you’ll get tails on either the first or the 2nd toss of the coin. Now you’ll use addition and subtraction:

You have 4 possible outcomes:

  1. Both coins land on heads.
  2. Both coins land on tails.
  3. The first coin lands on heads and the 2nd one on tails.
  4. The first coin lands on tails and the 2nd one on heads.

Of those 4 possibilities, only 1 of them doesn’t have tails as one of the results. So the probability of getting tails on either of the coins is 3/4, or 0.75, or 0.75%, or 1 to 3.

What Does this Have to do with Expected Value?

To find the expected value of a bet, you multiply the probability of winning the bet by the amount you stand to win. You multiply the probability of losing the bet by the amount you stand to win. Subtract one from the other, and you have the expected value of the bet.  Here’s an example:

You bet $1 that a coin will land on heads. If you win, you get $1, but the casino keeps a 10 cent commission. If you lose, the casino gets your dollar.

You have a 50% probability of winning 90 cents, which is +$0.45 in expected value.

You also have a 50% probability of losing $1, which is -$0.50 in expected value.

Total those, and you’ll see that the expected value of the bet is $-0.05.

If you make that bet under those conditions repeatedly for a long time, eventually, you’ll see an average loss of $0.05 per bet over time – even though it’s impossible to win or lose more than $1 or 90 cents at a time.  Another way to look at it is to look at all the potential outcomes and assume you get all of them. How much money do you have left afterward?

An Expected Value Using Roulette

Let’s use roulette as an example:

You place a single-number bet on the number 19.

There are 38 numbers on a roulette wheel, and only one of them is 19. This means the probability of winning this bet is 1/38.

This bet pays off at 35 to 1 odds. If you bet $1 and lose, you’re out a buck.

But if you win, you get $35 in winnings.

That sounds great, but look at what happens over 38 statistically perfect spins:

You win once for $35.

But you lose 37 times, at a dollar each, for a total loss of $37.

Your net loss is $2 over the 38 spins.

This means that you lost an average of $0.0526 per spin over those 38 spins.

And that’s the expected value of the bet, which is also the house edge.

Why Is It Important to Understand Expected Value?

It’s important because you should be an educated consumer no matter what you’re buying. In the case of a casino game, you’re paying for entertainment. You need to understand what that entertainment costs.

If you don’t understand expected value and the house edge, you have no idea what you’re up against.

You can’t compare one bet to another to decide which is better. (The better bet is always the one with the lower house edge.)

Understanding expected value is one of the first goals every casino gambler should have.

Laura Kirsten

Blogger and Casino Fan, originally from Argentina.

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