The Use of Mathematics in Gambling

You might not realize it now, but by the time you get through this article, you’ll see that math is everywhere in gambling.

It’s in the blackjack hands you play, the toss of the dice, and every single spin on your favorite slot machine.

And while mathematics is certainly important in gambling – you can use it to make smarter, more profitable bets – we’d wager that most people who read this won’t have much use for it.

We’ll explain why a little later on.

Before we get to that, though, we want to first talk about where and how math is used in gambling. Along the way – assuming you play poker, blackjack, or bet sports – we’ll show you how you can use this info to your advantage.

Let’s get to it.

Where Math Is Used in Gambling

Math is everywhere in gambling. But the most common math-type thing you’ll see, hear, or experience, is probability – the likelihood that something specific is going to happen.

Example of Probability in Gambling

Let’s use a Texas hold’em hand for example.

Let’s say that two Texas hold’em poker players have gone all-in pre-flop. One has aces and the other has kings. Here’s what the probability of each person winning looks like:

  • As Ad – 81.06%
  • Kh Kc – 18.55%
  • Tie: .38%

This number will vary a little bit, depending on the suits of the cards.

Now, the probability here will change as each round – the flop, turn, and river – is dealt. For example, say the flop is Jd 4d Qc. Here’s how the probability changes for each hand:

  • As Ad – 90.81%
  • Kh Kc – 9.19%
  • Tie: 0%

Big difference, yeah? Now let’s say the turn is the Ks. Here’s what the probability for each hand looks like:

  • As Ad – 13.64%
  • Kh Kc – 86.36%
  • Tie: 0%

Last card – the dealer flips over the 10s – giving the player with aces the win and the pot.

Does that make sense?
That’s a rough idea of what probability is, how it works, and what influences it.

And we’ll talk more about what influences gambling math here in a minute.

But probability of winning/losing exists for every single game or outcome in the casino. It can shift wildly from game to game, round to round, and in either direction (win/lose/tie) at any given time.

You can see 14 excellent probability examples in our blog post.

(Pot) Odds in Gambling

We also have odds in gambling.

Now, we’ve already talked about probability, which is pretty much the same thing – how likely something is to happen – or what the odds of a specific outcome are.

But odds can also express how much you will spend and how much you can receive if your bet wins.

If you’re given 9:1 odds on a football match, you’ll win 9x your investment if that bet wins. If you’re given 5:1 odds, then you’ll receive 5x your investment.

Knowing that, if you plan to make this bet and you knew that Sportsbook A is offering the 9x odds and that Sportsbook B is offering the 5x odds, then doesn’t it make more sense to take the 9:1 odds? Because the probability of winning doesn’t change, yet the odds you’re receiving are MUCH better.

You can also compare different kinds of odds, one being probability, to determine if your bet or play is profitable in the long run (we’ll talk about the long run here in a moment).

Take our AA vs KK hand above, for example. If you knew without a doubt – with 100% certainty – that your opponent has aces (while you’re holding kings), you’d know that you’re a 4:1 underdog.

That means that going all-in on the flop will be unprofitable unless you’re getting a “good price” to call the bet. In poker, this is often called pot odds. Let’s look at a quick example of pot odds so you can see how it works in practice.

Say there is $100 in the pot and it’s $50 to call. This means you’re getting 2:1 pot odds. If you do the math – where you make the call four times and lose, and one time and win – here’s what that will look like for you:

  • Call 4x and lose: -$200
  • Call 1x and win: $150

This means that you’re making a -EV (negative expectations) call at a loss of $50, which means this play loses money in the long run.

Your pot odds need to be better if you want to make a profitable call.

You already know that you’re a 4:1 underdog, so to keep this example short and simple, we’ll tell you that you need at least 4:1 on your money to break even. But we don’t play poker to break even; we play to win. So we need odds larger than 4:1 to make the call profitable.

So, let’s say the pot was $500 and still only $50 for you to call. Here’s how the math changes in your favor:

  • Call 4x and lose: -$200
  • Call 1x and win: $500

Over this small run, you can see that you’d spend $200 to win $500 for a $300 profit. That makes this a profitable and – depending on the exact situation – easy call.

Pot odds are more noticeable in poker, as you have the information in front of you (pot size, your stack, your potential hand, and an idea of your opponent’s hand). It’s easier to do the math.

But you won’t always be able to do the math for other casino games. More than that, though, most casino games outside of poker and sports betting are set up to favor the house, despite however good or skilled you may be.

That being said, there are other variables you can pay attention to, memorize, or otherwise look for/avoid, to stack the odds in your favor as much as possible.

What Influences Your Odds (Probability)?

Now let’s talk about what affects the odds of each game or bet you make in the casino. This is important because you can use these variables to find the best game or variant to play.

Here are some of the variables that influence probability:

  • Rules – You’ll notice this A TON in blackjack, as every casino has their own blackjack games and set of rules. For example, you may or may not be able to:
  • Split aces
  • Double down more than once
  • Split pairs more than once

And so on. Every single rule influences the odds you receive for the game you’re playing.

  • Number of Decks – This is also a common blackjack variable. The fewer decks used, the easier it is for you to know which cards are still available, which means your odds are better. The more decks in play, the harder this becomes – your odds are worse.
  • Payouts – We’ll use blackjack again because it’s such an easy example. But some games pay the standard 2:1 on blackjack, while others (that you should avoid) pay 6:5. This affects the house edge (your odds) in the game.
  • Number of Players – Let’s use a drawing or raffle, for example. If you put your name in a bowl, you stand a much better chance of winning – your odds are better – if there are 4 other players in the bowl compared to if there were 400 names in the bowl.

The all-in example above also applies. Aces are a bit more than 80% to win versus kings in a heads-up (two player) situation. But what if you added three more players, all with random hands? This will affect the probability of the aces winning like so:

  • As Ad – 45.64% | Tie: .34%
  • Kh Kc – 18.61% | Tie: .14%
  • Ac Tc – 7.67% | Tie: .34%
  • 8s 6d – 16.05% | Tie: .14%
  • Qc Jc – 11.7% | Tie: .14%

You see what we mean?

What’s crazy here is that the chances of the pocket kings winning hardly changed, while the pocket aces’ chances of winning was reduced by 35+%.

Not only that, but the 8-6 off-suit has an almost equal chance of winning the hand as the pocket kings do, and they have a better chance of winning than the QJ suited.

Okay, let’s look at one last example of how the number of players (or bettors) affects odds.

In sports betting, the lines you get (the odds you pay/receive) shift based on the bets made and what they’re made on. For example, you might be able to get -7 on a team/match, but if you wait 5 or 10 minutes, those odds might change to -7.5. This means you pay more for the same bet, which reduces your profit.

  • Timing – A good example of this are video poker progressive jackpots. Most times when you play, these the odds will favor the house. However, if you time it right – as Huck Seed did in 2011 – the jackpot will be large enough that you can have a small edge (< 1%). This means it’s profitable to play each hand while trying to hit the progressive.

Now, there’s no guarantee you’ll hit it, and you can deplete your bankroll before you do. But if you have the bankroll to cover the costs, you can earn a few bucks per hand…and several (hundred) thousand for the jackpot IF you get lucky.

  • Machine – Slot machines are a good example here. Some slots have better odds than others. The WizardOfOdds.com says that the simpler the machine, the better the payout. The odds are worse on the fancy machines with big signs.
  • Denomination – The WizardOfOdds.com also says that the smaller the denomination you use, the higher the casino edge will be. One example he gave is from a Clark County slot win from 2012 – if you spent $.01, the casino win percentage was 10.77%. But if you used $100, this dropped down to 4.73%.

It’s for this reason he advises that,

“it is better to play one coin per line on a 5-cent game than five coins per line on a 1-cent game.”

We also want to point out that sometimes the odds or house edge is really small. But so long as it’s in the casino’s favor, they will win…and over time they will win A LOT through volume (of hands/rounds played).

Other times it’s really big. Poker is a good example of this, especially the closer you are to the end of the hand. By then the odds are likely stacked more in favor of one player than the rest.

Variance in Gambling

We’ve looked at how math is used in gambling. And we’ve looked at the different variables that affect the probability of every game you play, or situation you’re in.

However, something that gets overlooked and is VERY important to remember is that there’s variance in every game or situation you play.

What is variance?

ThePokerBank.com puts it well when he says:

“Variance is the difference between how much money you expect to win on average over the long run and the results you are seeing in the short term.”

In other words, variance is the ups and downs you experience, and it, too, is math.

Poker is a great example of variance, as anyone who’s played the game has undoubtedly experienced a “bad beat” where they got it in with the best hand but still lost the pot.

Take the AA vs KK example from above. Aces are roughly 80% to win, so when you have aces and lose every once in a while – 1 out of 5 times – that’s perfectly normal. It’s expected. Poker players call that running at expectation.

However, if you lose 4 out of 5 times, that’s not in line with what’s expected – what the odds say should happen – that’s variance.

Where people get screwed up is that variance is about the short run, which is tiny compared to statistically relevant results over TONS – hundreds of thousands or millions – of hands or rounds. It’s perfectly normal to lose multiple times with aces (vs kings) over 100 or 1,000 hands. It’s frustrating, but normal.

But the closer you get to statistical relevance, the more the aces will win, and the closer they’ll get to that 81-ish% mark.

For another example, take betting on the outcome of flipping a coin. You know that, assuming a fair coin and everything else being equal, the outcome is 50% either way.

In the short run, though, the results can look like this:

  • Heads
  • Heads
  • Heads
  • Tails
  • Heads
  • Tails
  • Heads
  • Heads
  • Tails

And only after enough flips will it start to even out.

Let’s look at one more example: the Daniel Rainsong Challenge from the WizardOfOdds.com.

The Wizard of Odds issued a challenge that no betting system could beat the house advantage over the course of one billion hand computer simulations. He put up $20,000 against the $2,000 of the challenger.

Daniel Rainsong thought his system could beat the house because he had shown a profit of $8,215 over 33,000 hands of casino play with a starting bet of $1.

Now, as a side note, this is where people get screwed up. They see a “small” sample size of 33,000 with $8,000 in winnings and think that this is a legit and profitable betting system. But 33,000 hands aren’t enough.

You see this all the time in poker, too. People make plays based on short-term results, even though in the long run they’re not good plays.

Anyway, back to the story….

They settled on the rules and stakes, and then hit the button on the simulator. But they didn’t get through a billion hands. In fact, they didn’t even get close.

They stopped at 168,621 blackjack hands. This was the last point the betting system showed any profit. By the time they got to 10 million hands, the bankroll fell so deep into the hole that it was obvious the system would not work in the long run.

Rainsong forfeited the challenge.

The story above shows why sample size matters, and how long it might take for variance to run its course before the actual expectations (the probability) show.

That’s why it’s important to remember that almost no matter what casino game you play,
you’re destined to lose if you play it long enough.

And that’s why we believe that, for most people reading this, mathematics in gambling simply don’t matter.

Why Math Won’t Matter to the Majority

By this point, we’ve thrown a lot your way. We’ve talked about:

  • The types of math used in gambling.
  • How you use these odds to determine how much you’ll get paid.
  • How to use these odds to determine profitable bets.
  • The different variables that influence gambling math.
  • Variance in gambling, and how that impacts odds in the short run.

Now let’s talk about why nearly none of that will matter to most people reading this.

The reason why math won’t matter to most people is that nearly every game in the house is designed so that the player will lose.

Not only that, but (most) players know this and simply don’t care. They play these games anyway.

So, there’s not much sense in thinking about probabilities or odds, because if people truly cared about these and wanted to turn a profit, they’d never step foot in the casino in the first place.

Gambling is a form of entertainment. Most responsible people know this and go into the casino with an amount of money they’re comfortable losing, knowing that it’s the most likely outcome.

They’re fine with that so long as they’re being entertained.

Now, let’s say that you’re a little different. You still may want to play the occasional slot machine, but you’d rather be able to shift the math in your favor when/if it’s possible to do that. How would you go about doing that?

Here are some simple tips for you:

  • Learn which kinds of (slot) machines have the best odds. Chose to play those instead of the big fancy machines.
  • Learn basic video poker strategy. Take this one step further and – assuming you’re able to bet the max – learn when a video poker machine is +EV to play.
  • Learn basic strategy for any game before you sit down to play.
  • Learn basic odds and pot math to know when to chase a hand or fold in poker.
  • Learn how to compare rules, payouts, and sportsbook lines to ensure that you’re always getting the most money if/when you win.
  • Make sure to always get comps.

We have many resources on GamblingSites.com to show you how to accomplish each one. And for the majority of people reading this who are interested in improving the odds, this will be enough.

However, if you really want to shift the math in your favor, this is what you ought to do:

  • Play poker, blackjack, video poker, or bet sports.
  • Know when the best time to play video poker is.
  • Focus on and become great at one poker game. Really study strategy and focus on reducing mistakes.
  • Focus on one sport and become a consistently profitable (50%-60% win rate) sports bettor.
  • Learn how to count cards at blackjack.

This is MUCH easier said than done. You can play poker for the rest of your life and still not be one of the best players. And you can be one of the greats, but still have so much left to learn.

That goes for all the different games or betting opportunities, too.

There are pros/cons to each, too, such as the amount of time and money you’ll need to invest to become great. You also have the challenge of finding action, as weaker poker players will want to avoid you, and many casinos and sportsbooks will not want an advantage player taking them for a ride.

But those are the challenges you accept/face if you want to shift the mathematics in gambling into your favor.

Conclusion

Gambling math is important – even fun for some people – but unless you’re out to make a bunch of money, it’s not going to make a huge difference in the experience you have gambling.

Most people are going to play or gamble on whatever they want, regardless of the probabilities of them winning/losing, or how profitable an opportunity it may or may not be.

But mathematics is everywhere in gambling.

And if you take the time to at least get some of the basics down,
you can learn how to manipulate it and put it in your favor.

And that’s something anyone can do.