It’s important to understand that this term isn’t solely used in gambling. It originates from probability theory in mathematics, and is commonly used in economics.
Expected value is the mean average of a large number of independent, variable outcomes. We’ll try to translate that a bit. Expected value is a sort of predicted outcome of a variable, based on the average probability of each outcome. Let’s explain with a quick example.
Say you’re rolling a 6-sided die a thousand times. We’ll use dice because every outcome is equally likely, and there are six of them. If you keep rolling the dice, you’ll statistically eventually get the same outcome a roughly equal number of times.
If we add all of the results together and divide them by the number of rolls, we’ll probably get somewhere around 3.5. That’s the expected value of rolling dice. Every time you roll dice, the result will be 3.5 on average.
Keep in mind that the definition of expected value may slightly vary depending on who you ask. For example, in investment economics, the expected value (EV) is an anticipated average value for an investment in the future. It uses the same principles but for different purposes.
Importantly, expected value in the context of gambling and investments shows the profit or loss you’re about to see.
In other words, expected value in gambling shows how much your stake is going to change and not just the flat total.
Expected Value and House Edge
Now let’s go back to casino games and how any of this relates to them. You may have noticed that all this talk of probability ties in closely to Return to Player rates, or RTP. There’s a reason for that.
First of all, you have to understand that all casino games were designed to give players a negative expected value. That’s why they say that the house always wins. On average, every bet made at a casino will lose. That’s what luck-based gambling is actually all about – hoping for results that are decidedly not average.
Return to Player is basically an inverted, percentage representation of the expected value of a bet. Again, let’s translate.
By inverted, we mean that you’re representing how much of your stake you’ll keep, not how much your stake will change. Basically, RTP is your stake minus the expected value.
This leads us to the next conclusion: that the terms expected value and house edge are almost interchangeable. They show the same thing in a slightly different way.
Specifically, expected value in gambling does take into account the size of your stake. For instance, say you’re betting $10 in a game of European roulette. As we all know, it has an RTP of 97.30%, which means the house edge is 100-97.3=2.7%.
If you keep making the same bet with the same stake over and over again, you will theoretically lose $0.27 each time. Some bets will win, others will lose, but on average, you’re losing 27 cents per spin. Like we said – the house always wins.
How to Calculate Expected Value of Any Bet
So now you know the expected value of roulette bets. What about other games?
We’ll go through some of the most popular ones later. For now, we’ll give you a simple formula for calculating the expected value of bets at any moment.
Expected Value=((probability to win) x (profit)-(proability to lose) x (stake) )/Stake
That’s fairly easy to remember. And we can confirm it using the roulette numbers from above. Let’s say you’re making an Even Money bet, which has a probability of success of 48.60%, which is 48.6/100 or 0.486. You can find the probabilities for all roulette bets in our guide.
Your stake is $10. The profit is also $10 – Even Money pays 1:1, after all.
Expected Value=( (48.6/100 x 10)-(51.3/100 x 10) )/10
Expected Value=( (0.486 x 10)-(0.513 x 10) )/10
Expected Value=(4.86-5.13 )/10
In other worlds, you’re expected to lose just under 3 cents with every roulette spin.
Before you ask – there’s no real way to get a positive expected value in gambling. As we said, the games were designed so that it’s always negative. The only real exception is counting cards in blackjack, which boils down to waiting until the deck offers positive expected value. That’s a story for another time, though.
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