Overview
It's often said that you can't win at a casino, that even if you get lucky and win a few times you'll end up losing money in the long run.
In this post, we take an analytical look at whether this statement is true and why.
Expected Returns - Single Die Game
First, let’s start with a simple game where you a single 6-sided die is thrown, and you win if you can correctly guess the number it lands on.
If it’s a fair die (not loaded in any way) and thrown on a flat surface, the probability of it landing on any one number is equal to all other possible numbers. So since there are 6 sides (i.e. 6 possible outcomes), the probability of each number is \({1 \over 6}\).
Now let’s say the payout is 3 to 1, meaning you win three times your bet if you guess correctly but lose your bet if you’re wrong. What are your expected returns on this game?
Since the probability of you winning is \({1 \over 6}\) and the payout ratio is 3, your expected returns for a $1 bet is the payout multiplied by the probability of winning, subtracted by the bet amount multiplied by the probability of losing:
\[\eqalign{
ER &= {Payout \times P(Win) - Bet \times P(Lose)} \cr
&= {(3 \times {1 \over 6}) - (1 \times {5 \over 6})} \cr
&= -0.33 \cr
&= -33\%
}\]
This means you can expect to lose a third of your money in the long term, if you keep playing this game with $1 bets.
Would you play such a game? I most certainly wouldn’t!
What if the payout were to be increased to 5 to 1? Now the expected returns is
\[{5 \over 6} - {5 \over 6} = 0\]
Meaning you will break even playing this game over the long term.
Expected Returns - Roulette
Next, let’s calculate what the expected returns at a roulette table looks like. To make it simple, we’ll start with the single number bet.
At first glance, it looks like a slightly skewed payout where you pick a number out of 36 to bet on, and if the ball lands on your chosen number you’ll get 35 times your bet as payout.
But wait. If you observe the roulette, there are more than 36 numbers! The most common European system has an additional 0, the French/American system has both 0 and 00, while some rarer systems even have another 000.
Which means that on a European roulette the expected returns is
\[{35 \over 37} - {36 \over 37} = -0.027 = -2.7\%\]
The expected returns on a French/American roulette is even worse, at
\[{35 \over 38} - {37 \over 38} = -0.053 = -5.3\%\]
How about other bet types, such as Even/Odd, dozen, column and such? The existence of the 0 slot ensures that the probability of winning is slightly lower than the probability of losing.
For example, let’s look at an Even bet where you win if the ball lands on an even number and you lose if it lands on and odd number or zero. The payout is 1 to 1, meaning you get your money back if you win.
But since the odds of winning are \({16 \over 37}\) verses the odds of losing at \({17 \over 37}\), the expected returns is
\[{16 \over 37} - {17 \over 37} = -0.027 = -2.7\%\]
House Edge
This is negative skew from a theoretically fair payout is called the "house edge", being the mathematical guarantee that the casino (also called the "house") will always earn money over the long run.
Different casino games have different mechanisms of ensuring that the house edge exists, with some having higher edge than others.
For more complicated games such as Texas Hold’em, you need to calculate the probability of every single possible outcome and compare against the payout/loss at each of them. When you sum all of these up, you can see what the expected return of playing such a game is.
The house edge can be lower than 1% for certain card-based table games, while it can be more than 10% for slot machines.
Conclusion
Let’s imagine casino with a roulette section of 20 European tables, where there are on average 500 patrons playing over the course of each day and on average they bet a total of $500 over the duration of their play. Due to the house edge of 2.7%, the casino can expect to earn an average of $6,750 every day from this section alone, or $2.46 million a year!
If you think about all the different games available at any decent-sized casino, each of them with a distinct house edge, it’s easy to see why casinos are such lucrative businesses.
You might want to think about this the next time you’re on holiday, see a casino and get tempted to try your luck at winning a quick buck.