Seven Illustrations of Gambling Math in Use

By Team on July 29, 2022

The mathematics of gambling never ceases to fascinate. Gambling wouldn't even exist without the field of mathematics known as "probability;" at the very least, we wouldn't be able to discuss it rationally.

Fair wagers are uncommon. Almost often, one side enjoys an advantage over the other. Being an informed gambler requires being able to identify that edge. This post begins with an explanation of what probability is and how it is calculated, then moves on to seven instances of its use in real-world scenarios.

The goal of probability is to quantify the likelihood that specific events will occur. I'll refer to those items in this post as "events" for convenience. Probability is probably a term you use subconsciously while discussing potential outcomes.

Probability is most often expressed in terms of percentages, particularly when watching the evening news. The likelihood that there will be rain is indicated by the meteorologist's statement that there is a 50% risk of thunderstorms tomorrow. The majority of people also grasp that 50% indicates that it will rain 50% of the time and not 50% of the time.

Simply said, probability is a number that indicates how likely an event is to occur. And that figure is always in the range between 0 and 1. Nothing will ever happen if the probability is 0. Something will always occur if the probability is 1, which is also equal to 100%.

While percentages are one way to communicate probabilities, this is not the only option. As a fraction, you can also express it. 50 percent is equivalent to one-half.

A probability can also be written as a decimal. The percentage of 50% is equal to 0.5.

Additionally, probability can be written in odds format. In this situation, 50% is equivalent to a 1:1 or even chance.

Each of those probabilistic expressions has a place in various contexts. When comparing the reward of a bet with the odds of winning that bet, it is very helpful to express a probability as odds.

Actually, probability calculation is quite straightforward. You compare the number of possible outcomes for a single event to the number of possible outcomes overall. The fraction is constructed by placing the single event at the top and the total number of possible events at the bottom. Of course, if you have even the slightest math background, you are aware that division can be used to convert a fraction to a decimal or percentage.

Depending on whether you want to know if several events will occur or if you want to know the chances of a specific number of events occurring, you can either multiply or add when calculating the probability of multiple occurrences.

The terms "and" and "or" are the ones to pay attention to in this situation.

You multiply the probabilities of each occurrence if you want to determine the likelihood that both events A and event B will occur.

You add the probabilities of each event if you want to determine the likelihood that either event A or event B will occur.

The examples that follow will demonstrate how frequently these probability computations take place in the gaming industry.

1. Roulette Math

A straightforward game like roulette is an excellent way to see how probability works. 38 outcomes are conceivable on an American roulette wheel, which has the numbers 0, 00, and 1-36. The zero and zero are both green. The other numbers are split equally between black and red.

You may compute the likelihood of virtually any outcome or combination of outcomes using this information. These probabilities can be compared to the payouts for the wager to determine whether one side has an advantage and, if so, how great of an advantage.

Let's start by considering some of the outside bets, which are some of the more popular wagers in roulette. These wagers can be placed on red/black, high/low, or odd/even. Each pays out at the same odds. You wager $1 on one of these outcomes, and if you win, you get back $1.

That seems like a reasonable bet at first, but upon closer inspection, it becomes clear that the house has a considerable advantage.

This is why:

Consider placing a black bet. There are 20 numbers on the wheel that are not black, compared to 18 black numbers. Two more numbers are green, and there are 18 red numbers. Only 18 of the 38 potential outcomes, therefore, result in your bet winning.

The probability is now 18/38. The best way to interpret this wager is probably to translate it to a percentage, which is 47.37 percent.

Therefore, the casino will prevail in this wager 52.63 percent of the time while you do so the remaining 47.25 percent of the time. It is obvious that if you play this game long enough, the casino will eventually take home the entire pot.

Even better, you can determine how much of each wager the casino will ultimately win; this amount is known as the house advantage.

How to do it is as follows:

Assume you place 100 bets and get the predicted outcomes based on math. (That never occurs in real life, but if you play for long enough, the outcomes will begin to resemble one another.)

In this situation, you will gain $47.37 but lose $52.63. This results in a net loss of $5.26 ($52.63 - $47.37).

You lost an average of 5.26 percent of each wager because you made a total bet of $100 on those 100 wagers.

The house edge is that.

It turns out that the house edge for all roulette bets is that (except for one).

The green 0 and the green 00 are sorts of where the house has an advantage. If those numbers weren't on the wheel, the rewards for all of the bets at the table would provide neither side an advantage.

They are, nevertheless, at the wheel. And that really does make a difference.

2. A Coin Toss Mathematical Basis

A coin toss is an even more basic illustration of probability in action. Although they could, the majority of people don't actually bet on coin toss results. Additionally, one side may or may not have an advantage over the other side depending on the payment structure.

Here is the calculation in its most straightforward form. If you flip a coin, you want to know the likelihood of getting heads. Given that there are only 2 possible outcomes and only one of them is a head, your probability is 12, or 50%.

Flipping a coin is used when you want both sides to have an equal chance of winning. This is how, for instance, it is decided who will kick off during a football game.

It's important to note that calling heads or tails first has no advantages. I don't think in psychic phenomena, yet the probability is the same. I've never come across any proof that someone possesses any sort of precognitive power that would increase their chances of correctly guessing the result of a coin toss.

Let's attempt a more intriguing computation, though. Let's imagine we want to know the likelihood of receiving two consecutive heads. You therefore need to know both the likelihood of obtaining heads on the first flip and the likelihood of getting heads on the second flip.

Recall that I indicated previously that we multiply if the problem contains the word "and." In this instance, multiplying 1 by 1 results in the number "1" Alternatively, if we name it 0.5 X 0.5, we get 0.25. 25 percent can be stated in either of those ways.

Looking at the total number of outcomes after tossing a coin twice in a succession is another way to approach this:

  • 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 of a coin toss. Let’s say you’re running a backroom 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. Math in Poker

I could discuss poker math for the remainder of this piece. But I'll make an effort to keep it to this single bullet point.

Anyone who has played poker at all understands that you have an equal probability of obtaining a better hand than I do. The 52-card deck from which we both draw cards is the same one.

The key is in what you do with those cards afterward.

Consider a scenario in which you are playing 5 Card Draw and are dealt a hand that has 4 cards that would make a flush. In the hopes of drawing to that flush, you'll discard a card.

How likely is it that you will be successful?

The deck still has 47 cards. Nine of them have the suit you require. (Each suit consists of 13 cards, four of which are in your hand already.) You have a 9/47, or 19.1 percent, chance of receiving the card you require. That amounts to 20% or about 1 in 5.

Calculate the amount of money in the pot that must be present for you to profitably call a bet if you consider that you must win this hand in order to win the pot.

Assume there is $10 in the pot and you must pay $1 to stay in the game and receive the additional card. If you triumph, your odds of winning a 4 to 1 draw are 10 to 1. Nearly 80% of the time, you'll lose, but the 20% of wins that you do have will more than make up for your losses and net you a nice profit.

In fact, let's perform the identical computation as above, supposing you perform this action 100 times in a row. You'll earn $190.10 while losing $80.90, giving you a $109.20 profit. These are very good pot odds.

The payout wouldn't be large enough to make this a worthwhile wager, though, if there were only $3 in the pot and it cost you $1 to enter. You would still lose $80.10, but you would only gain $57.30, resulting in a $22.50 loss overall.

Of course, other probabilities would need to be considered in an actual poker game. To scare your opponents out of the pot, for instance, you might raise in this scenario. When you use this strategy, you must determine the likelihood that it will be successful. To your predicted value, you can add that.

Here, it becomes crucial to read other players. Some people believe that reading people entails predicting their actions 100 percent of the time.

In actuality, though, you can only make educated assumptions about their propensity to act. Your strategy will be significantly altered if you believe that your opponent will call your bluff 50% of the time.

4. Video Poker Math

Video poker is somewhat similar to both poker and slot machines, but it is more similar to itself than either of those things. But the majority of the arithmetic is comparable to that used in conventional poker. The distinction is that when you obtain a specific hand, you know exactly how much money you will win. Concerning what your rivals have, you need not fret.

As an illustration, if you hold a pair of jacks in a poker game and your opponent also has a pair of jacks, you might tie and share the pot.

On the other hand, with a Jacks or Better video poker game, you always win even money when you get a pair of jacks or higher. Furthermore, neither a pair of queens nor a pair of kings will result in a larger payoff. Despite the fact that in a genuine poker game there is a clear hierarchy among those 3 hands, for the sake of these rewards, all 3 hands are treated equally.

Draw poker is the game's foundation, so each time you receive a hand, you get to choose which cards to keep and which to discard. In order to determine which choice has the best expected value, you assess the likelihood that you will hold a certain set of hands with their associated payouts.

As an illustration, consider the following

A royal flush is the best potential hand you can obtain in the majority of video poker games, and it pays out at an incredible 800 to 1. (I'm assuming you're betting the maximum number of coins; otherwise, the payout is only 250 to 1). However, you should never play for fewer coins than the maximum.)

With a pair of jacks or higher, though, you can win at even odds. That obviously has a considerably lesser payout.

But what if you had to pick between those two possibilities? If you hold the ace, king, queen, and jack of hearts, for example, you are in good shape. The king of spades, though, is your fifth card.

You are dealt two kings. You can keep it with a 100% chance of receiving an even-money payout.

Alternately, you might discard the king of spades and attempt to get the royal flush. There is a slimmer than 2% chance that one of the 47 remaining cards will be the one that completes your hand.

Over a hundred flawless iterations, what happens?

If the odds are 98 to 1, you lose. On the other hand, twice as many coins are awarded. This is 1600-98 or 1502. This is 15.02 for each wager you earned when multiplied by 100 wagers.

In the alternative scenario, you receive 100 wins overall but only receive 100 coins.

Would you want to win $1 per bet on average or $15 per bet?

The chance that you could draw to a different random winning hand is obviously ignored in this example, although that option has roughly similar odds with both choices. Just let's suppose that it balances out.

The chances of hitting your hand decrease significantly if you only have three cards to a royal flush. 2.2% times 2.2% equals 0.04 percent. With those odds, you'll need much more than an 800 to 1 reward to justify that choice.

The decision you must make, however, has a higher predicted value than any of the others, regardless of the first card you are dealt.

This expected value is calculated by considering all of the potential actions in that circumstance and the likelihood that each one will produce a specific payment 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

Blackjack is the subject of my preferred gambling math. This classy game is also one of the few in the casino where a good player might get an advantage. The game has a memory, which makes it more intriguing.

This is what I mean:

On each turn of the roulette wheel, the chances are the same. The probabilities of what will happen on the next spin are unaffected by the results of previous spins. Each time the wheel is spun, there are 38 potential outcomes, and each one has an equal chance of occurring.

The chances would vary with each spin, though, if you removed a slot from the wheel after it was hit.

Here is an illustration:

You place a bet on black. That wager has an 18/38 chance of success.

You succeed. The ball is kept in that slot by the croupier (the roulette dealer), preventing further landings on that spot.

You place a new wager on black. Since one of the possibilities has been eliminated, the chance of winning this time is merely 17/38.

Every time a card is dealt in blackjack, this is exactly what occurs. In the following rounds, only 51 of the 52 possibilities are still accessible.

Until the dealer deals another round of cards, this continues.

It goes without saying that the chances are always the same in a game with a continuous shuffler.

However, the majority of games are still dealt by hand without the aid of a machine. In these games, you may roughly keep track of the cards dealt and increase your bets when your chances of earning more money are better.

That works like this:

The payout for a "natural," sometimes known as "blackjack," is 3 to 2. A two-card hand with a 21-point total is that. Aces, which count as 11, and the 10, J, Q, and K, each of which counts as 10, are the only card values that can produce such a hand. All other card values, such as J, Q, and K, do not.

Blackjacks are impossible to obtain if a deck's aces are all gone. You simply cannot accomplish this.

The likelihood of getting a blackjack also decreases each time a 10 is handed.

However, the chances shift a little bit in favor of the player each time a lower-ranked card, such as a 2, 3, 4, 5, or 6, is dealt.

Because of this, a card counter will employ a system to roughly track the ratio of high cards to low cards. High cards are given a score of 1, whereas poor cards are given a score of 1. The counter is aware that he has a higher than usual probability of receiving the 3 to 2 payoff if and when the count increases significantly on the positive side. As a result, he increases his wagers. More wagers are placed by him as the count rises.

As soon as the count is zero or negative, he reduces his wager.

But those are the fundamentals of card counting. There is much more to it. They also have a mathematical foundation.

7. Math for Sports Betting

The majority of bookies don't only say you have to risk $110 to win $100; they also do other things. By granting or taking away points, they can also disadvantage teams. A bet on either side will have a 50/50 chance of winning thanks to this handicapping. A 50/50 proposition is advantageous for the bookmaker but unprofitable for the player because these sports wagers don't pay out at even odds.

However, when they establish the odds, the bookies aren't always accurate. Additionally, they don't always leave the lines in their current state. Equal action on both sides of an event is what a bookmaker aims for. They take this action in order to use the winning wagers as collateral for the losses. The loser's preferred method of profit-making is with that additional $10 bet.

However, what if not an equal number of bets are placed on either side?

The line is typically moved by bookmakers to encourage movement on the other side. It's typically preferable to bet against the public, according to shrewd sports bettors who are familiar with how the industry operates.

Here is a demonstration of how this operates:

With a 7-point advantage against the Dallas Cowboys, the Washington Redskins are playing. Therefore, the bookmaker deducts 7 from the Redskins' final score before paying out a wager on them.

The Cowboys don't collect nearly as many wagers as they anticipate despite the fact that they established this line early in the week. The line is then shifted to 7.5 in an effort to spur more activity on the other side. The public is typically mistaken in situations like this, thus a wise bettor will place a wager in opposition to it.

What happens to the necessary winning percentage to even break even is the really fascinating effect of the vigorish on a sports bettor. You will experience a loss of funds if you are half right and half wrong. Half of the time, you lose $110, while the other half, you only gain $100.

You can break even or even turn a small profit if you can place your bets on the correct side slightly more than 53% of the time. You're well on your way to become a top-tier sports betting if you can surpass the 55 percent mark and start getting close to the 60 percent mark. With that kind of winning percentage, you can earn six figures a year, but you'll need to have enough cash on hand to last through any losing streaks you could experience.

There will inevitably be losing streaks in the short run. A game of chance by its own nature is like that. Furthermore, the handicappers who work for the bookmakers are nearly always correct. You must have the ability to spot successful circumstances if you want to win money betting on sports. To do this requires frequently outsmarting handicappers and bookmakers.

A topic that never gets old is finding value when you bet on sports.


These seven examples show that getting a fair bet is uncommon, as you can see. Every so often, someone gains an advantage. Comparing the payouts for winning bets and the winning odds for each side allows you to determine who has the advantage and by how much.

The advantage held by casinos over customers is constant. The double up bet in video poker and the odds bet in craps are the only bets I can think of in a Las Vegas casino that have reasonable chances. However, there are a few wagers that the player has an edge over, but they are the outliers, not the rule, and you can find them occasionally in Las Vegas casinos.

There's not a lot you can do to even out the chances when playing games like slots, craps, and roulette. I'm doubtful of claims made by certain people that they can influence the result of a dice roll.

However, you might be able to gain a slight advantage over the casino if you're a skilled player at video poker or blackjack. The majority of casinos, though, won't let you keep playing if you're counting cards while playing blackjack. And they are now fairly proficient at catching players who have an advantage.

Sports bettors and skilled poker players can skew the odds in their favor, but they still need to be talented enough to overcome a simulated house advantage. The rake, or percentage of each pot charged by the cardroom that hosts the poker games as table rental, is a standard practice in the game of poker. You must wager $110 to win $100 in sports betting. The vigorish refers to the additional $10 you must stake on each wager.

You'll appreciate it better if you grasp the math involved in the game and your bets, regardless of the type of betting you choose to engage in. To examine the math involved in gambling, I create posts.

Although trying to get an advantage is worthwhile, doing so is impossible if you don't have at least a basic understanding of gambling math. My early success was greatly aided by seeing it in action.