Understanding the House Edge in Roulette and Beyond

How to crunch the house edge?

So, what’s the house edge, and how do we calculate it? How are the values in the table above arrived at? The game and the version you play always determine the house edge. It’s based on each outcome's odds and the bets' fixed payout ratios. Here’s how the house edge is calculated.

$\mathrm{House\; Edge}=\left|\frac{\mathrm{Expectation}}{\mathrm{Bet} \mathrm{Amount}}\right|\times 100\%$

Put simply, the house edge is the expectation divided by your bet, then multiplied by 100. Alright, let’s now take a closer look at the formula.

$\mathrm{House\; Edge}=\left|\frac{(-\mathrm{loss\; amount}\times \mathrm{odds\; of\; loss})+(\mathrm{win\; amount}\times \mathrm{odds\; of\; win})}{\mathrm{Bet\; Amount}}\right|\times 100\%$

In this case, the meaning of the metrics is as follows:

• Loss amount – This is the amount lost when your bet fails (the size of your stake)
• Odds of loss – This is the probability of your bet losing in the first place (the ratio between winning and losing outcomes possible)
• Win amount – This is the amount you stand to win (equal to the bet amount and multiplied by the payout ratio, i.e., 35:1, and so on)
• Odds of winning – This is the chance of winning with a certain outcome (once again based on the comparison of possible winning and losing outcomes)

Alright, but what if we wanted to get even more precise – well, there is a solution for that as well!

$\mathrm{House\; Edge}=\left|\frac{(-\mathrm{bet\; amount}\times \mathrm{odds\; of\; loss})+(\mathrm{bet\; payout}\times \mathrm{bet\; amount}\times \mathrm{odds\; of\; win})}{\mathrm{Bet\; Amount}}\right|\times 100\%$

Right, this is great, but what if we took a real betting situation? Let’s bet 100 AUD on Red, where the payout is 1:1. The formula then changes to this:

$\mathrm{House\; Edge}=\left|\frac{(-100\times 19/37)+(1\times 100\times 18/37)}{100}\right|\times 100\%=\left|\frac{(-51.3) + \left(48.6\right)}{100}\right|\times 100\%$

$\mathrm{House\; Edge}=|-0.027|\times 100\%=2.7\%$

We have a 2.70% house edge. As you can see, all the numbers match. But hold on – isn’t that just a trick? Isn’t this house edge 2.70% because we used even-money bets? This is a good question. So, let’s change the bet – place 10 AUD on a straight bet with a 35:1 payout and see what happens:

$\mathrm{House\; Edge}=\left|\frac{(-10\times 36/37)+(35\times 10\times 1/37)}{10}\right|\times 100\%=\left|\frac{(-9.73)+\left(9.46\right)}{10}\right|\times 100\%$

$\mathrm{House\; Edge}=|-0.027|\times 100\%=2.7\%$

We just got the same house edge! OK, but aren’t there situations where the bet type will change the house edge? You are right – there are, but they aren’t too many. Let’s bet 5 AUD on an 8:1 payout in American roulette, and see what we get as a result:

$\mathrm{House\; Edge}=\left|\frac{(-5\times 34/38)+(8\times 5\times 4/38)}{5}\right|\times 100\%=\left|\frac{(-4.4736)+\left(4.2105\right)}{5}\right|\times 100\%$

$\mathrm{House\; Edge}=|-0.0526|\times 100\%=5.26\%$

Alright, in this case, the house edge is 5.26%. A quick look at the table for American roulette, and voila – the house edge is indeed 5.26%. As you can see, the house edge is fixed as an immutable constant, but as you have already noted, there may be some exceptions. Let’s take a look at these exceptions specifically next.

Exceptions that prove the rule

Even-money bets in French roulette

French Roulette has a pretty interesting twist in its rules. The rule is called La Partage, and it applies to even-money bets. If you lose because of zero, the La Partage rule kicks in, and what it does is split the bet and returns half of it. Because of this, the house edge for even-money bets is not 2.70% but rather – 1.35%. Let’s try this in practice and see how it works.

We are going to place a 5 AUD bet on Red. We will now miss because of a 0. Here is the formula that kicks in:

$\mathrm{House\; Edge}=\left|\frac{(-5\times 18/37)+(1\times 5\times 18/37)+(-5/2\times 1/37)}{5}\right|\times 100\%=\left|\frac{(-2.4324)+\left(2.4324\right)+(-0.0675)}{5}\right|\times 100\%$

$\mathrm{House\; Edge}=|-0.0135|\times 100\%=1.35\%$

Basket bet and American roulette

Up next, we have another exception to the rule. The five-number bet in American roulette, better known as the basket bet. It consists of 0, 00, 1, 2, and 3. The bet we place will again be 5 AUD, and it pays 6:1. The formula we follow is:

$\mathrm{House\; Edge}=\left|\frac{(-5\times 33/38)+(6\times 5\times 5/38)}{5}\right|\times 100\%=\left|\frac{(-4.3421)+\left(3.9473\right)}{5}\right|\times 100\%$

$\mathrm{House\; Edge}=|-0.0789|\times 100\%=7.89\%$

Once again, the house edge is slightly different than what we expected – 7.89%. This exception only applies to this selection in particular.

What about other versions?

Alright, we have analyzed the house edge on particular selections and bets, but what about other versions of roulette? We have prepared several notable exceptions here!

How does the house edge make money for casinos?

Right, so let’s now focus on the house edge and how the casino will end up making a profit because of that. If you have a starting bankroll of 100 AUD, you will forfeit the house edge percentage as a loss to the casino, which is the casino’s profit. Let’s take a look at a few examples right now.

• If you play European roulette, the casino will profit 2.70 AUD for every 100 AUD you bet – a 2.70% house edge
• If you play American roulette, the casino will profit 5.26 AUD for every 100 AUD you bet – a 5.26% house edge
• If you play French roulette and bet on even-money bets, you will forfeit AUD 1.35 for every 100 AUD you bet, and the house edge is set at 1.35%

As you can see, there is clear-cut internal logic that is easy to follow and understand. However, it’s important to understand that these figures only work in the long term. In the short term, it’s not necessarily guaranteed that for every 100 AUD you will lose an amount equal to the X house edge.

100 AUD is still too little to determine that, and the best way to get the house edge is to play a minimum of around 1,000 rounds. The important thing, however, is that the longer you play, the higher the chance that the casino will turn a profit for you.

This brings us to our next point – the gambler’s fallacy. The gambler’s fallacy will make you mistakenly believe that if you just stick around for a little longer, you will end up in the black.

While it’s technically true that you can always use a betting system, these systems do have merits. However, they won’t magically disappear from the house edge. While these systems are fantastic for bankroll management, they will still cause you to lose in the long run.