Most of the math taught in school lacks a practical application.
That’s why I think gambling math terms and concepts are easier to learn. The practical applications couldn’t be more obvious. After all, what’s more practical than money?
When it comes to most kinds of casino gambling, you’re mostly going to be concerned with the hourly cost of your entertainment.
But other kinds of gamblers—advantage gamblers—are interested in understanding the math well enough to make a living from their wagers.
In either case, understanding a handful of terms and concepts related to the subject can go a long way toward helping you achieve your goals, regardless of what they are.
What it means: “Probability” refers to the likelihood that a particular event will actually happen. An event’s probability is always a number between 0 and 1, with a probability
of 0 meaning that the event will NEVER happen, and a probability of 1 meaning that the event will ALWAYS happen. You can also calculate the probability of multiple events happening.
Probabilities can be expressed in multiple ways:
As a fraction, like ½.
As a percentage, like 50%.
As a decimal, like 0.5.
As odds, like 1 to 1.
How to calculate a probability: An event’s probability is most easily understood as a fraction or a division problem. The numerator (the number on top of the fraction) is the
number of ways in which a particular event can happen. The denominator (the number on the bottom of the fraction) is the total number of possible events, including the desired event and all the
When calculating a probability as a division problem, the desired event is the dividend (the number being divided), and the total number of possible events is the divisor (the number being
divided by). The quotient (the solution to the division problem) is the probability.
Here’s a simple example:
You want to know the probability of getting heads when you flip a coin. There are two possible events—heads or tails. Only one of them is heads. That means the probability of getting heads when
flipping a normal coin is ½. Of course, the probability of getting tails is also ½.
Here’s a slightly more complicated example:
You want to know the probability of rolling a 6 on a standard 6-sided die. There are 6 possible outcomes, but only one of them is a 6. The probability of rolling a 6 is 1/6. The probability of
rolling any other number is 5/6.
You can easily convert a probability stated as a fraction to a decimal by simply dividing.
½ = 0.5
1/6 = 0.1667
You can also easily convert a decimal into a percentage by multiplying the decimal by 100.
½ = 0.5 = 50%
1/6 = 0.1667 = 16.67%
You can also express these numbers in odds form. You’re just comparing the number of outcomes that are and aren’t the desired event.
½ = 1 to 1
1/6 = 5 to 1
Expressing a probability as odds can be useful when comparing the payoff of a bet with the odds of winning that bet.
You can also calculate the probabilities for multiple events. Depending on whether you’re using the word “and” or the word “or”, you’ll either add or multiply.
Here’s an example:
You want to know the probability of rolling a 1, 2, OR a 3 on a 6-sided die. Since you’re using the word “or”, you’ll add the probability of each of those results in order to get the probability
of getting any one of them.
In this case, you have a 1/6 chance of getting each of those numbers, so the equation looks like this:
1/6 + 1/6 + 1/6 = 3/6, which reduces to ½.
You might also want to know the probability of rolling a 1 on 3 dice at the same time, or successively. To pull this off, you have to roll a 1 on the first die, AND a 1 on the 2nd die,
AND another 1 on the 3rd die. Since you’re using the word “and”, you’ll multiply. The equation looks like this:
1/6 X 1/6 X 1/6 = 1/216.
All gambling math questions begin with the calculation of probability.
What it means: The word “percentage” is derived from the expression “per cent”. “Cent” means 100, so “per” cent means “per 100”. And that’s exactly what a percentage is—a ratio
expressed as a number per 100.
Most people are pretty familiar with percentages, at least to some extent, because percentages are used in the world of personal finance pretty often.
Here’s an example:
The most common advice about savings is that you should save 10% of what you earn. That means if you earn $50,000 a year, you should be saving $5000.
Here’s another example:
Suppose you take out a loan with annual percentage interest rate of 10%. Let’s say you borrowed $10,000. You’ll pay $1000 a year in interest on top of the principal.
But neither of those examples have anything to do with gambling.
Here’s a gambling-related example related to percentages:
You’re playing in a poker game where they rake 5% of the pot. (The “rake” is how the poker room makes money hosting the game.) You’re in a pot worth $100, but you only win $95, because 5% of the
pot goes to the house.
How to calculate the percentage of something: It’s easy to calculate the percentage of something. You just multiply the percentage by the something.
In the above example, I talked about how you would save $5000 per year if you earned $50,000. I calculated that by multiplying 10% by $50,000.
That might be easier to understand if you convert the percentage to a decimal. You do that by dividing by 100. (Remember that we multiply decimals by 100 in order to convert them to a percentage.
To convert them back, you divide.)
10% divided by 100 is 0.1. It’s easy to understand 0.1 X $50,000 = $5000.
The applications related to gambling will become clearer as you familiarize yourself with the rest of the terms on this page.
Odds are usually expressed as the odds against something happening. For example, if you’re rolling a single 6-sided die, your odds of rolling a 1 are 5 to 1. There are 5 ways to NOT roll a 1, but
only a single way to roll that one.
How odds inform your betting decisions: But suppose you made a bet with someone, and they offered to pay you 7 to 1 if you correctly predict the roll of the die. Since 7 to 1 is
better than 5 to 1, this represents a profitable betting opportunity.
On the other hand, if someone offered you 4 to 1 if you predict the next roll correctly, the betting opportunity is still profitable—but not for you!
Here’s how you can look at it. Suppose you rolled the die 6 times and got statistically perfect results. That means you’d lose the bet 5 times and win once. If you bet $1 every time, you’d lose
$5. But you’d win $7 on the bet you won, which leaves you with a profit of $2.
Using the other example, if you bet $1 every time, you’d still lose $5 on your losing bets. But on the bet you win, you’ll only win $4. That extra dollar goes to the other person.
Of course, everyone knows that we don’t see statistically perfect results in the short term.
But in the long run, over the course of hundreds or thousands of trials, the actual results will start to resemble the statistically expected results.
What it means: “Expected value” refers to what a bet is worth to you mathematically over the long run. If the expected value is positive, then if you make that bet repeatedly
over the long run, you should see a profit. If it’s negative, you should see a loss. In the short term, anything can happen.
Here’s how to calculate expected value: This one seems harder to calculate than some of the others, but it’s really not.
To calculate the expected value of a bet, you multiply the probability of winning by the amount you’ll win.
Then you multiply the probability of losing by the amount you’ll lose.
You add the results of those 2 calculations together to get your expected win or loss.
Here’s an example:
You have a friend who’s crazy. He offers to pay you $2 if you correctly guess the result of the next coin toss, but if you lose, you only have to pay him $1.
You have a 50% chance of winning $2. The product of those 2 numbers is $1.
You also have a 50% chance of losing $1. The product of those 2 numbers is -$0.50. (It’s negative because that’s the amount you’ll lose, not win.)
$1 + -$0.50 = $0.50.
So your expected value on that bet is $0.50, and it’s a positive expectation bet, because you expect to win.
Here’s another example from an actual casino game:
You’re playing American roulette at a casino in Las Vegas. You bet $1 on a single number.
That bet pays off at 35 to 1, but the odds of winning it are 37 to 1.
You stand to win $35 if you win, but the percentage chance of winning that is 2.63%.
$35 X 2.63% = $0.92
On the other hand, you have a 97.37% chance of losing the bet.
-$1 X 97.37% = -$0.97
The expected value of that bet is $0.92 + -$0.97, or -$0.05.
In the short run, you might win early and often. But if you play roulette long enough, you’ll eventually tend toward the trend of losing a nickel on every dollar you bet.
This is how the casino makes its money on every bet on every casino game, by the way. The payoff for the bet is always less than the odds of winning that bet. Over the enormous number of bets per
hour taken in by a casino over the course of 24 hours, the casino is almost guaranteed a profit.
And that’s why casinos stay in business.
Positive expected value for the casino and negative expected value for the player.
If you read the example above about expected value in roulette, you’ll notice that your expected to lose about a nickel for every dollar you bet.
Expressed as a percentage, that’s 5%. That’s only an approximation, though—the actual house edge on roulette is 5.26%.
How do you calculate the house edge: You convert the expected loss on a $100 bet into a percentage.
Average Hourly Cost
What it means: Many casino gamblers look at the average hourly cost of playing a casino game in order to get a rough gauge for how much entertainment they’re getting for their
The casinos use this same information to decide how to maximize the amount of profit they make from the square footage in the casino.
How to calculate the hourly cost of playing a casino game: This is another simple calculation. You only need to know 3 pieces of information, and you multiply all 3 of them to
get the average hourly cost:
Average bet size
Number of bets per hour
The house edge
Here’s an example:
You’re playing a slot machine for $1 per spin. (That’s your average bet size.) You’re an average player, so you’re making 600 spins per hour, which means you’re putting $600 into action per hour.
(That’s your number of bets per hour multiplied by the average size of each bet.) This particular slot machine has a 6% house edge, so you can expect to lose $36 per hour playing it.
Of course, most slot machines don’t list the house edge, but the house edge for a lot of other games in the casino is known because of the nature of the game.
Here’s an example of that:
We know that in American roulette, almost every bet (all but one, in fact), has a house edge of 5.26%. An average roulette player might be wagering $5 per bet and making 50 bets per hour. That
means he’s putting $250 per hour into action, and he expects to lose 5.26% of that, or $13.15.
Even though the house edge in this example is roughly the same, the game is SO much slower that the hourly cost to play is dramatically lower.
Anyone who ignores blackjack instead of roulette ought to re-consider. The difference in terms of how much entertainment you get for your dollar is dramatic.
Return to Player or Payback Percentage
What it means: This is the flip side of the house edge, and it’s used mostly when referring to slot machines or video poker games. This is the percentage of each bet that, over
time, is statistically expected to be returned to the player in the form of winnings.
To calculate the return to player (RTP) or payback percentage, you simply subtract the house edge from 100%. If a slot machine has a house edge of 6%, the return to player is 94%.
You can do this calculation in reverse to convert the payback percentage (which is just another word for return to player) into the house edge. If a game has a 94% return to player, you know that
the house edge is 6%.
The house edge and the return to player always amount to 100% when added together.
What it means: “Advantage gambling” refers to gambling only when you have a positive expected value. In other words, if you expect to lose money on a bet over a statistically
significant period of time, you don’t make the bet. You only place bets when you have an edge.
Examples of advantage gambling: Most casino games offer no opportunity for a player to get an edge over the house. The math just doesn’t add up. But there are exceptions. They
generally require a lot of skill and dedication that amounts to stubbornness.
Card counting in blackjack is one example. By tracking the ratio of high cards to low cards in the deck, a skilled blackjack player can raise her bets when the odds are in her favor. By doing
this, she makes up for all the negative expectation bets and then some.
Expert play at video poker combined with taking advantage of slots club promotions is another advantage gambling technique.
Here’s how that works:
The return to player on a Jacks or Better video poker game with a 9/6 pay table is 99.54%, so the house edge is only 0.46%. (This assumes you’re making the correct strategic decisions, by the
But suppose you join the slots club and get 0.3% of your expected losses back. You’re still facing a net house edge of 0.16%, so this isn’t advantage gambling yet.
But suppose that for 2 hours every Monday, Wednesday, and Friday, the casino offers double or triple rewards to their slots club members. Now you’re looking at getting back 0.6% or 0.9%.
Since 0.6% and 0.9% are both greater than 0.46%, you’re not in a positive expectation situation.
Other ways to get a statistical edge in gambling include becoming an expert horse race handicapper, becoming an expert sports bettor and handicapper, and becoming an expert poker player. None of
those are casino games, per se, but they are opportunities to engage in some advantage gambling.
One distinction is important to make, here, too. Advantage gambling generally refers to techniques that are within the rules of the game. If you were to mark the cards during a poker game, you’re
a cheater, not an advantage gambler.
Variance or Standard Deviation
What it means: Variance and standard deviation are what we call results that don’t conform to the statistical expectation. In the short run, anything can happen. You could win
the first single number bet on the roulette wheel and quit while you’re ahead. That would make you a profit even though you made a negative expectation bet.
Mathematicians measure how far from the expected results your actual results are by a number called standard deviation. But for practical purposes, you just need to understand that what most
people call luck is just another word for standard deviation.
What it means: Vigorish refers to the commission that bookmakers charge for taking your action. Most bookies require you to risk $110 to win $100. That extra $10 represents the
Vigorish is often abbreviated as “vig”. Sometimes people refer to the house edge of a casino game as the vig, but that’s a sort of imprecise use of the term.
If you want to make a profit at sports betting (or any other game), you have to find a way to win often enough to compensate for the vig and still make a profit.
Being an educated gambler begins with understanding some of the math behind the game. Luckily, most of this math can be easily understood if you learn what a few terms mean and how to calculate
them. I’ve listed the main ones above in this post. Once you’ve mastered these concepts, you’re pretty much all set.