The math behind gambling is endlessly fascinating. In fact, without the branch of mathematics called “probability”, we wouldn’t even have gambling—or at least we wouldn’t be able to talk about it intelligently.
Few bets are fair bets. One side almost always has an edge over the other. Being able to determine that edge is a critical part of being an educated gambler. This post starts with an overview of what probability is and how it’s calculated, then it continues with 7 examples of how it’s used in practical applications.
Probability concerns itself with measuring how likely it is that certain things will happen. For purposes of this post, I’ll call those things “events”. You probably use probability to talk about possible events without even knowing it.
Probably the most common expression of probability happens with percentages, especially when you’re watching the nightly news. When the meteorologist says that there’s a 50% chance of thunderstorms tomorrow, she’s telling you what the probability is that there will be rain. And most people understand that 50% means that half the time it’s going to rain, and half the time it’s not.
A probability is just a number that describes how likely an event is. And that number is always a number between 0 and 1. Something with a probability of 0 won’t ever happen. Something with a probability of 1 (which is also 100%) will always happen.
You can express probabilities as percentages, but that’s not the only way to express a probability. You can also express it as a fraction. 50% is the same thing as ½.
You can also express a probability as a decimal. 50% is the same thing as 0.5.
Probability can also be expressed in odds format. In this case, 50% is the same thing as 1 to 1, or even odds.
Each of those ways of expressing probability is useful in different situations. Stating a probability as odds is especially useful when comparing the payoff of a bet with the odds of winning that bet.
Calculating probability is actually pretty simple, too. For a single event, you look at the number of ways that event can happen versus how many ways things might turn out total. You put the single event on top of the fraction, and you put the total number of potential events as the bottom of the fraction. Of course, if you have any math experience at all, you know that you can use division to turn a fraction into a decimal or a percentage.
If you want to calculate the probability of multiple events, you either multiply or add depending on whether you want to know if multiple events will happen or if you want to know the odds of a certain number of events happening.
The key words to look for in such a problem are “and” and “or”.
If you want to know the probability that event A will happen AND event B will happen, you multiply the probability of each.
If you want to know the probability that event A will happen OR event B will happen, you add the probability of each.
The following examples will show how these probability calculations happen time and again in the gambling world.
1. Roulette Math
Roulette is a simple game, and it’s a great example of probability in action. An American roulette wheel has 38 possible events, numbered 0, 00, and 1-36. The 0 and the 00 are green. Half of the other numbers are black, and half of them are red.
With this information, you can calculate the probability of just about any outcome or combination of outcomes. You can compare those probabilities with the payoffs for the bet to see if one side has an edge, and if so, how much that edge is.
Let’s start by thinking about some of the more common bets in roulette—the outside bets. These bets are on odd/even, high/low, or red/black. They all pay out at even odds. You bet $1 on one of these outcomes, you win $1 if you win.
At first glance, that sounds like a fair enough bet, but when you look at these bets a little more closely, the house has a distinct advantage.
Suppose you bet on black. There are 18 numbers on the wheel that are black, but there are 20 numbers on the wheel that are not. (18 of the numbers are red, and 2 more numbers are green.) So out of 38 possible outcomes, only 18 of them win your bet.
That makes the probability 18/38. It’s probably easiest to understand this bet by converting it into a percentage, 47.37%.
So 52.63% of the time, the casino will win this bet, and the rest of the time, you will. It’s clear to see how if you play this game long enough, eventually the casino will win all your money.
You can even calculate the amount of each bet the casino will win over the long run—this number is called the house edge.
Here’s how you do it:
Assume that you make 100 bets and that you see the mathematically expected results. (That never happens in real life, but if you play long enough, the actual results will start to resemble the expected results.)
In this case, you will win $47.37, but you’ll lose $52.63. That’s a net loss of $52.63 – $47.37, or $5.26.
Since you bet $100 on those 100 wagers, you lost an average of 5.26% of each bet.
And that’s the house edge.
As it turns out, that’s the house edge for all the bets at the roulette table (except for one).
In a sense, the green 0 and the green 00 are where the house gets its edge. The payouts for all the bets on the table would offer neither side an edge if those numbers weren’t on the wheel.
But they ARE on the wheel. And that makes all the difference.
2. The Math Behind a Coin Toss
An even simpler example of probability in action is a coin toss. Most people don’t actually place wagers on the outcomes of a coin toss, but they could. And depending on the payout structure, one side might or might not have an edge over the other side.
Here’s the simplest version of this calculation. You want to know the probability that you’ll get heads on a coin toss. Since there are 2 potential events, and since only 1 of them is heads, your probability is ½, or 50%.
In cases where you want both sides to have an even shot at winning something, you’ll flip a coin. This is how they determine who gets to kick off during a football game, for example.
I should point out that there’s no advantage to being the one to call heads or tails. The probability is the same, and I don’t believe in psychic phenomena. I’ve never seen any evidence that anyone has any kind of precognitive ability that would improve their chances of predicting the outcome of a coin toss.
But let’s try a more interesting calculation. Let’s say we want to know the probability of getting heads twice in a row. That means you want to know the probability of getting heads on the first flip AND the probability of getting heads on the second flip.
Remember I said earlier that if we’re using the word “and” in the problem, we multiply. In this case, we’re multiplying ½ by ½, which is ¼. Or we could call it 0.5 X 0.5 and get 0.25. Either of those ways can be expressed as 25%.
Another way to look at this is to look at the total number of outcomes when you toss a coin twice in a row:
- You could get heads on the first toss and heads on the second toss.
- You could get tails on the first toss and tails on the second toss.
- You could get heads on the first toss and tails on the second toss.
- You could get tails on the first and heads on the second toss.
Those are literally the only 4 outcomes, but only 1 of them is the outcome you were solving for. That’s ¼, or 25%, which is what we’d determined earlier.
Suppose you wanted to create a simple gambling game based on the outcome a coin toss. Let’s say you’re running a back room casino in a bar or something.
You might have a game where you toss a coin, and so does the dealer. If you get heads and the dealer gets tails, you win. If the dealer gets tails, and you get heads, then the dealer wins.
But if you both get heads or both get tails, you have to put up another coin in order to get to toss the coins again.
The catch is that the dealer does NOT have to put up another coin. If you win this second toss, you win a coin, but if you lose it, you lose both coins that you put up.
It’s pretty clear in this example how the casino has an edge, right?
3. Poker Math
I could spend the rest of this post talking about poker math. But I’ll try to limit it to just this bullet point.
Anyone who knows anything about poker knows that you have just as good a chance of getting a better hand as I do. We’re both getting cards from the same 52 card deck, after all.
It’s what you do with those cards after that make a difference.
Let’s suppose that you’re playing 5 card draw and you’re dealt a hand with 4 cards to a flush in it. You’re going to discard a card and hope to draw to that flush.
What is the probability that you will succeed?
There are 47 cards left in the deck. 9 of them are of the suit you need. (There are 13 cards in each suit, and 4 of them are already in your hand.) So your probability of getting the card you need is 9/47, or 19.1%. That’s almost 1 in 5, or 20%.
If you assume that you have to make this hand in order to win the pot, you can calculate how much money needs to be in the pot in order for you to profitably calla bet.
Let’s suppose that there is $10 in the pot, and it costs you $1 to stay in and draw that extra card. If you win, you’ll win 10 to 1 on a 4 to 1 draw. You’ll lose almost 80% of the time, but you’ll win so much when you do win that it will make up for it and give you a tidy profit.
In fact, let’s do the same calculation we did above, where we assume that you do this 100 times in a row. You’ll lose $80.90, but you’ll win $190.10, for a profit of $109.20. These are excellent pot odds.
On the other hand, if there were only $3 in the pot, and it cost you $1 to get in, you wouldn’t get a big enough payout to make this a profitable bet. You’d still lose $80.10, but you’d only win $57.30, for a net loss of $22.50.
Of course, in a real poker game, you’d have other probabilities to take into account. For example, you might raise in this situation, hoping to scare your opponents out of the pot. You have to estimate the probability that this tactic will work when you try this. You can add that to your expected value.
This is where reading other players becomes important. Some people think that reading people is all about gauging what they’re going to do 100% of the time.
But the reality is that you make educated guesses about their likelihood of doing something. If you estimate that your opponent will fold to your bluff 50% of the time, then that makes a big difference to your strategy.
4. Video Poker Math
Video poker is a little bit like poker and a little bit like slot machines, but it’s like nothing so much as it’s like itself. Most of the math, though, is similar to the math of traditional poker. The difference is that you have an exact payoff you can expect when you achieve a certain hand. You don’t have to worry about what your opponents have.
For example, if you have a pair of jacks in a poker game, and your opponent also has a pair of jacks, you could wind up in a situation where you tie and split the pot.
But in a Jacks or Better video poker game, you get paid even odds any and every time you get a pair of jacks or higher. And you don’t get a higher payout for a pair of queens or a pair of kings. For purposes of these payouts, all 3 hands are the same, even though there’s a clear hierarchy among those 3 hands in a real poker game.
Video poker is based on draw poker, so every time you get a hand, you get to decide which cards to keep and which ones to throw away. You compare the probability of making certain hands with their payoffs in order to decide which decision has the best expected value.
Here’s an example:
The best possible hand you can get in most video poker games is a royal flush, which pays off at a whopping 800 to 1. (I’m assuming you’re making the max coin bet—if you don’t, you’re only getting a 250 to 1 payoff. But you should never play for less than max coins.)
But you can win even odds with a pair of jacks or higher. That’s clearly a much lower payoff.
But suppose you have to choose between those 2 options? Let’s say you have the ace of hearts, the king of hearts, the queen of hearts, and the jack of hearts. But your 5th card is the king of spades.
You have a pair of kings. You can keep that and have a 100% chance of getting an even money payoff.
Or you can throw away the king of spades and try to get the royal flush. Only 1 card of the 47 remaining cards will make your hand, which is a slightly better than 2% chance of success.
What happens over 100 perfect iterations?
98 times you lose your bet. But twice you get 800 coins. That’s 1600-98, or 1502. Divided by 100 bets, that’s 15.02 per bet that you won.
In the other case, you win 100 times, but you only win 100 coins total.
Would you rather average $15 in winnings per bet, or $1 in winnings per bet?
Of course, this example ignores the possibility that you could draw to another random winning hand, but that has a more or less equal probability with both decisions. We’ll just assume that it evens out.
On the other hand, if you only had 3 cards to a royal flush, the odds of hitting your hand get much smaller. 2% X 2% is 0.04%. With odds like that, you’ll need a lot more than an 800 to 1 payoff to make that decision worthwhile.
But no matter what hand you are dealt initially, you have one decision which has a higher expected value than any of the others.
That expected value is determined by looking at all the possible moves in that situation and the likelihood that each of them will result in a particular payoff amount.
5. Craps Math
Craps is an interesting exercise in probability because it’s a great example of a bell curve. That’s when some results happen so seldom that the drawing of the curve is low on either end, but the odds of the results in the middle happening are much higher.
Here are the possible outcomes when rolling a pair of dice:
2 – 1 +1 – Only one possible way of getting this total.
3 – 2+1 or 1+2 – Only 2 possible ways of getting this total.
4 – 3+1, 2+2, or 1+3 – Only 3 possible ways of getting this total.
5 – 4+1, 3+2, 2+3, or 1+4 – Only 4 possible ways of getting this total.
6- 5+1, 4+2, 3+3, 2+4, 1+5 – Only 5 possible ways of getting this total.
7 – 6+1, 5+2, 4+3, 3+4, 2+5, 1+6 – Only 6 possible ways of getting this total.
8 – 6+2, 5+3, 4+4. 3+5, 2+6 – Only 5 possible ways of getting this total.
9 – 6+3, 5+4, 4+5, or 3+6 – Only 4 possible ways of getting this total.
10 – 6+4, 5+5, or 4+6 – Only 3 possible ways of getting this total.
11 – 6+5 or 5 +6 – Only 2 possible ways of getting this total.
12 – 6+6 – Only one possible way of getting this total.
You only have 11 possible totals, but you have a total of 36 different outcomes.
Knowing this, you can divide the number of ways of achieving each total by 36 in order to determine the probability of getting that total.
So getting a total of 2 or 12 has a probability of 1/36.
3 or 11 has a probability of 2/36, or 1/18.
4 or 10 has a probability of 3/36, or 1/12.
5 or 9 has a probability of 4/36 or 1/9.
6 or 8 has a probability of 5/36.
7 has a probability of 6/36, or 1/6.
So your most likely outcome is a total of 7, but that still only happens 1 time out of 6.
But you can bet on any of these totals at various times in the game. You can compare the payoffs on these bets with the odds of winning to determine the house edge on each of those bets.
For example, you can make a place bet on any 8 or any 6 and get a payoff of 7 to 6 if you win. But the odds of winning that bet are 5/36. That can be converted into a percentage, and we can calculate the house edge for that bet. The odds of winning this bet are 13.89%.
Place this bet 100 times, and you will win 13.88 bets with winnings of $1.17 each time (7 to 6). That’s $16.24 in winnings. But you lose 86.12 times, losing $1 each time, for losses of $86.12. You’ve lost way much more than you’ve won over these 100 bets–$69.88. That makes the house edge 6.99% on this bet, which is almost 7%. That’s worse than roulette with its 5.26% edge.
Luckily, many of the bets on the craps table have a much lower house edge.
6. Blackjack Math
My favorite kind of gambling math relates to blackjack. It’s such an elegant game, and it’s also one of the only casino games where a skilled player can get an edge. What’s so interesting about the game is that it has a memory.
Here’s what I mean:
When you play roulette, the odds are the same on every spin of the wheel. The outcome of one spin has no effect on the odds of the outcome of the next spin. There are 38 possibilities every time you spin the wheel, and each of them is equally as likely as the others.
But if you eliminated a slot on the wheel once it got hit, you’d wind up with odds that changed on every spin.
Here’s an example:
You bet on black. The probability of winning that bet is 18/38.
You win. The croupier (the roulette dealer) leaves the ball in that slot, so it can’t be landed on again.
You bet on black again. This time the probability of winning is only 17/38, because one of the options has been removed.
This is exactly what happens every time a card is dealt in blackjack. One of the 52 options is no longer available to be dealt in subsequent rounds.
This continues until the dealer reshuffles the pack of cards.
Of course, in a game with a continuous shuffling machine, the odds stay the same no matter what.
But most games are still dealt without the benefit of such a machine. In these games, you can keep rough track of which cards have been dealt and raise your bets when you have a better chance of winning more money.
Here’s how that works:
A “natural”, or a “blackjack”, pays off at 3 to 2. That’s a 2 card hand that totals 21. There are only 2 values of cards which can result in such a hand—the aces, which count as 11, and the 10, J, Q, and K, each of which counts as 10.
If all of the aces in a deck are gone, it’s impossible to get a blackjack. You just can’t do it.
Every time a 10 gets dealt, your chances of getting a blackjack decrease, too.
But at the same time, every time a lower ranked card gets deal, like a 2, 3, 4, 5, or 6, the odds improve a little bit in the player’s favor.
So a card counter will use a system to keep rough track of the ratio of high cards to low cards. They count the low cards as +1 and the high cards as -1. If and when the count gets high on the positive side, the counter knows he has a better than average chance of getting that 3 to 2 payout. So he raises his bets accordingly. The higher the count, the more he bets.
He lowers his bet when the count is 0 or negative.
There’s a lot more to counting cards than that, but those are the basics. And they’re rooted in math.
7. Sports Betting Math
Most bookmakers require you to risk $110 in order to win $100, but that’s not all they do. They also handicap teams by giving them points or taking them away. The goal of this handicapping is to make a bet on either side a 50/50 proposition. Since these sports bets don’t pay off at even odds, a 50/50 proposition is profitable for the bookmaker but not the player.
But the bookmakers aren’t always right when they set the lines. And they don’t always leave the lines the way they are. A bookmaker’s goal is to get an equal amount of action on either side of an event. They do this so that they can pay off the winning bets with the losers’ money. That extra $10 that the losers bet is how they prefer to make their profit.
But what if they don’t get an equal amount of bets on each side?
Most bookmakers move the line in order to stimulate action on the other side. Sharp sports bettors—those who know how the business work—know that it’s usually best to bet against the public.
Here’s an example of how this works:
The Washington Redskins are playing the Dallas Cowboys, and they’re favored by 7 points. This means that before paying off a bet on the Redskins, the bookmaker subtracts 7 from their score.
They set this line early in the week, but they don’t get nearly as many bets on the Cowboys as they expect. So they move the line to 7.5, which is meant to encourage more action on the other side. A smart bettor is going to bet against the public in this situation, because the public is usually wrong.
The really interesting effect of the vigorish on a sports bettor is what it does to the required winning percentage just to break even. If you’re right 50% of the time and wrong 50% of the time, you’ll lose money. You’re losing $110 half the time, and you’re only winning $100 the other half the time.
If you can bet on the right side a little over 53% of the time, you can break even and even make a tiny profit. If you can get over 55% and start nearing 60%, you’re on your way to becoming a world class sports bettor. You can make 6 figures a year with a win rate like that, but you need to have enough money in your bankroll to be able to weather any losing streaks you might run into.
Losing streaks in the short term are inevitable, too. That’s just the nature of a game of chance. Also, the handicappers who work for the bookmakers are almost always right. In order to make a profit betting at sports, you have to be adept at finding profitable situations. This means outthinking the handicappers and the bookmakers most of the time.
Finding value in sports betting is an endlessly interesting topic.
As you can see from these 7 examples, it’s unusual that anyone ever gets a fair bet. Someone almost always gets an edge. Figuring out who has the edge and by how much is just a matter of comparing the odds of winning for each side and the payouts for winning those bets.
Casinos always have an edge over the players. I can only think of 2 bets in a Las Vegas casino which offer fair odds—the double up bet in video poker and the odds bet in craps. But you can find occasional bets in Vegas casinos where the player has an edge, but these are the exceptions, not the rules.
When you’re playing games like slots, craps, and roulette, there’s really not much you can do to even out the odds. Some people claim that they can affect the outcome of a roll of the dice, but I’m skeptical.
On the other hand, if you’re a skillful blackjack player or a skillful video poker player, you might be able to get a small edge over the casino. If you’re counting cards as a blackjack player, most casinos will refuse to let you continue to play, though. And they’re pretty good at catching advantage players now.
Skilled poker players and sports bettors can get the odds in their favor, but they still have to be skillful enough to overcome a house edge of sorts. In poker games, the cardroom hosting the games charges a percentage of each pot as rent for the table—this is called the rake. When betting on sports, you have to bet $110 to win $100. That extra $10 you have to risk on every bet is called the vigorish.
But no matter what betting activity you choose, you’ll enjoy it more if you have a clear understanding of the math behind the game and your bets. That’s why I write posts examining the math behind gambling.
It’s worth it to try to get an edge, but it’s impossible to get an edge if you don’t have at least a rudimentary understanding of gambling math. Seeing it in action helped me a lot when I got started.
And if you have any aspirations of gambling professionally, understanding these examples is a must.