Understanding Expected Value in Casino Games

Every time we walk into a casino or a gambling game of any sorts, we dream about winning big. If the game is fair, there is always a chance that will happen, but also a chance we will end up losing. We depend on luck for the outcome, but luck is really just a physical manifestation of the mathematics at work.

Every game of chance, from a simple coin toss to roulette, is based on mathematics. Every time you flip a coin it will land on either heads or tails. In other games, the chances and odds are more complicated, but still perfectly calculable.

While we can never say with certainty how we will do in a certain gambling session, we can definitely have a good idea of how we will do over the long run if we play a certain game of chance. This expectation is called “expected value” by the experts and it determines just how much we will be winning or losing over the course of a lifetime playing the game.

Return to Player (RTP) and House Edge

Each gambling game that is played against the house comes with a certain advantage for the house. This advantage is called the house edge and it is the difference between the total percentage of money returned to the player by the game and the overall amount of bets made.

Return to player (RTP) percentage is the one stat that is always displayed on electronic games and well-known in card games like blackjack or baccarat. This percentage signifies the percentage of all money wagered on the game that will be paid back through various winnings.

In a game of blackjack, for example, players can come close to breaking even and leave the house with just a 0.5% edge, if they apply the correct strategy. Other games, such as slots, can give the house as much as 10% or even 20% edge, which means that the long-term return of these games will be very negative for the player.

The expected value of the game has everything to do with the RTP of the game in question. If you are playing a game that has a 90% RTP (and a 10% house edge), you are expecting to lose 10% of your bets in the long run. If you are betting $10 per spin, you will be losing $1 in every spin. Of course, the variance of the game will see you win on many spins and lose on others, but the overall result will be such that you will be losing a dollar every time you press the spin button.

For that reason, the only way to gamble responsibly is to play games that have the best possible RTP and the lowest house edge. This means playing games like video poker or blackjack and applying the best possible strategies each time. This may not sound like too much fun, but it is the only way to limit your losses.

How is Expected Value Calculated?

The expected value and house edge of most casino games and other gambling games is usually well-known. However, if you don’t believe the casinos or even the regulators and you want to find out for yourself, you can calculate the winning odds and expected value of every game by using simple mathematical formulas.

The more complex your betting pattern, the more mathematics is involved in calculating the expected value. However, to make things simple and calculate for a single bet, you will need to use the following formula:

Expected Value (EV) = (probability to win) * (total profit) – (probability to lose)

Let us use this in a simple example of betting red/black in the game of roulette, which we already know will come out to 2.7%, just like any other roulette bet:

Expected Value (EV) = (18:37) * (1) – (19:37)

Expected Value (EV) = 0.4864864864864865 – 0.5135135135135

Expected Value (EV) = 0.270270270270270135

The same formula can be used to calculated the expected value in any other game of chance if you know the exact odds of each bet you are making. Of course, if the odds are approximate like in sports betting, your result will be approximate as well.

RTP and Expected Value in Various Casino Games

Each game you can find in a traditional casino comes with a negative expected value. The only way for the casinos to make money, in the long run, is to make the games favour them ahead of the players. The negative expected value and the RTP of less than 100% is what makes players lose over the long run.

However, the games are not all the same, and despite what players might believe, there is every difference between choosing different games on the casino floor. You can find the full list of RTPs for different games in the Auscasinos RTP guide, but here are the RTPs of a few most popular casino games:

• Slots: Between 80% and 95%
• European Roulette: 97.3%
• American Roulette: 94.74%
• Blackjack: 99.5%
• Video Poker: 99% (depends on version)
• Caribbean Stud Poker: 94.78%

As you can see on this short list of casino games, they all come with a negative expected value. That means you can expect to lose some money every time you play these game on average. However, the closer the number is to 0, the more likely you are to win on a given session. The further it is from 0, the more chance there is you will end up losing some money.

The expected value and RTP should really be one of the first things you look at in a game. A casino game that’s very fun but comes with a very bad RTP like slots should be avoided. The negative RTP quickly adds up and players are left with nearly no chance of winning even on a given day, let alone the long run.

Keep in mind that there are also several gambling games, namely video poker machines, that can be played with a positive expected value. Full pay machines such as Deuces Wild and Joker Poker have RTPs of over 100% when played optimally. If you can find these machines and learn the way to play them right, you could actually beat the house in its own game.

How Expected Value Manifests in Real World

If you think about a very simple game of heads or tails, you should start to understand how gambling works. In a neutral setting, heads or tails has an expected value of 0 as there are two betting options and a payout of exactly 1:1. Provided the coin is not fixed, the chance to win is exactly 50%.

Now, go ahead and toss a coin 10 times. Chances are you will not get exactly 5 heads and 5 tails. In some sets of 10 tosses, you may even end up getting all 10 heads or 10 tails. This phenomenon is known in gambling as variance and it depends on the standard deviation statistic. The standard deviation is very high in games with many possible outcomes and lower in games with few possible outcomes. The closer a game is to 0 expected value, the lower the deviation will be.

In games like slots that sometimes give the house a 20% edge, the standard deviation can also be very high. This means players will have sessions where they will keep losing and lose way more than 20% of the money they wager. On the other hand, they will also have big winning sessions and the end result will be a loss of 20% of the money wagered over thousands of spins.

The thing to remember is that a little luck can actually ride you and keep you positive for a long time in games with a low house edge. However, when you play games that give the casino a significant advantage, any winnings streak you do find will be short-lived.