Explaining House Edge: The Ins and Outs

How to figure out the house edge in roulette?

If you are keen on figuring out the roulette edge yourself, then there is a quick way to see how this works exactly. The house edge will remain immutable and be based on metrics such as the expected outcome and the bet amount multiplied by a fixed value. Here is an example of what formula is used to calculate the house edge exactly:

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

In other words, the house edge is the result of the absolute value of the expectation divided by the amount that is being bet. To illustrate this, we will now examine the formula in more detail.

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

To help you understand, we have broken down the formula even further.

• Loss amount: This is what you lose when your bet doesn’t hit the chosen pocket or combo.
• Odds of loss: The chance your bet loses—calculated by comparing losing outcomes to all possible ones.
• Win amount: What you win if your bet hits. It’s your original bet multiplied by the payout rate.
• Odds of winning: The chance your bet wins, based on how many winning outcomes there are out of the total possibilities.

We can discuss the payout ratios in even more detail, helping you better understand their fixed values.

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

It would make a lot of sense to stop our theoretical approach here and focus on a practical example that happens with every spin of the roulette wheel. We will use a single-zero roulette here. Our bet amount will be 100 CAD on Black. As we probably know, Black is an even-money bet, meaning that its ratio would be 1:1. Here is what this looks like as a mathematical representation:

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

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

Alright, as you can see, everything we have put down now produces the house edge for single-zero roulette, which is 2.70%, but some of you may say – well, this is simply because you bet on even-money bets. This is a great question. Let’s change the bets and see what happens, and if the house edge for a straight-up bet, which pays 35:1, would be the same.

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

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

It seems uncanny, but once again, the formula has arrived at the very same conclusion. The house edge of European roulette, regardless of the bet you place, is still 2.70%. Alright, but what if we place a 5 CAD bet on a corner and even change up the version to American roulette? The payout here is 8:1.

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

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

In this case, the house edge is 5.26% and you may be tempted to say “a-ha, I told you there was something off,” only we are playing a double zero roulette and as per the table we shared at the beginning of the article, it’s evident that the house edge for this version is exactly that. However, there are exceptions to the house edge rule! Let’s address them head-on!

Exceptions: Are there any?

Even-money bets and French roulette

Right, so when it comes to French roulette, there is one key distinction that you may notice. These are the so-called La Partage and En Prison rules that allow you to either keep half of your lost wager or, when betting on zero, have another spin. The exception applies only to this game version and only in those instances when the losing number is zero. However, casinos rarely use the En Prison rule; La Partage is more common. The house edge is thus reduced to 1.35% instead of 2.70%.

Alright, but what does this look like when you consider the context of a mathematical formula? We have put together a quick example:

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

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

Basket bet and American roulette

Another exception to the house edge rule is the so-called basket bet, or the five-numbered bet in American roulette, where the house edge is up to 7.89%. Here, the house edge indeed changes because of the different bet type:

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

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

Other versions of roulette and house edges

How do casinos actually make money from the house edge?

Alright, we have touched on this already, but it’s worth going into a little more detail as to how the casinos turn a profit, leveraging the house edge to their advantage. In fact, the house edge is the very definition of “an advantage,” as we said earlier. As a player, you should think of this metric in the following way:

• In European roulette, the casino will win 2.70 CAD in the long run
• In American roulette, the casino will win 5.26 CAD when you play
• In French roulette, the casino may only win 1.35 CAD, but it still wins

This calculation is usually estimated at over 100 CAD, although this is still small for statistical accuracy. The more you play, the more the statistics start to favor the casino, which always has the upper hand. Yes, there is room for deviations – lucky spins and big wins, but the casino will win as long as you keep playing, as most people do.

Another important thing to note is the gambler’s fallacy. You should remind yourself not to fall victim to this mindset, which could falsely have you believe that the longer you stick around, the higher your chances are of turning around your fortune. This is not true. The longer you stay, the more likely you are to lose.

Naturally, there are betting strategies, and yes, they will help you maintain a healthy bankroll for longer, but even these strategies will eventually run out of luck. No strategy can somehow negate the simple fact that the house edge is always there, no matter what casino game you pick, and that it works for the casino.