How to calculate the house edge

Last update:  August 2019

In a separate article we saw how the house edge is the casino's average profit per bet, in percentage terms.  In this article we'll see how to calculate it.  The main point is to see that when you win, the casino always pays you less than true odds.  That's their built-in advantage over you, and why you can never be a long-term winner.

We'll start with coin flips, because they're easy to understand.  We'll bet $1 on the flip of a coin, winning on heads and losing on tails.  Here's how that looks:

There's no advantage to the casino in this game, which is why you'd never see this game in a casino.  So let's casino-ize it:  In this version, you get paid only 90¢ when you win:

Ta-da, house edge for the casino of 5%.  If you thought it should be 10%, remember, the 10¢ penalty kicks in only when you win, which is half of the flips.

Casino games are all based on the same principle, paying you less than the true odds of winning.  The games are just more complicated than our coin flip game, both to make them more interesting, and to make it less obvious that the odds are stacked against you.  Sure, we know the games are stacked against us, but if you can't easily see the inequality then you're more likely to play.  After all, how attractive did the 90¢ coin flip game seem to you?  Probably almost no one would play such a game.  But they'll play casino games, even though they're based on the exact same concept.

Let's look at the house edge in roulette.  There are 18 reds, 18 blacks, and two greens.  So if you bet on red, you have an 18/38 chance of winning.  If the casino were paying fair odds, they'd pay you the inverse, or 38/18, minus 1 (1 representing your bet, which you get back).  So, a fair payout on a winning bet would be3818 –1818 =2018.  That means on an $18 bet, they should pay us $20.  Let's see how that looks, with an $18 bet on red.

But, of course, the casino won't pay you fair odds.  Instead of paying you 20/18 on your win, they pay you only even money.  Let's see how that looks:

But, as we know, the casino always pays us less than fair odds.  They're not gonna pay us (38/24)-1 = 0.5833 of our bet, they're gonna pay us only 0.5 times our bet.  That is, we're gonna get 0 on the losing column, and 1 on the winning column, which averages to 0.5.  Let's see it in action:

Related articles:

• House edge for various games
• Figuring your expected loss
• Figure how much you can afford to bet
• All about betting systems
• Don't focus on the house edge for the lottery