When you are playing any casino game, you are dealing with a degree of unpredictability, even if it is a game that involves skill. But a lot of casino games have purely random outcomes: slots, roulette and keno are a few examples.
We talk a lot about randomness with respect to these games and why you cannot control their outcomes or give yourself an edge. But have you ever stopped and wondered what exactly it means for something to be “random” in the first place? What is randomness?
Discussions About Randomness Can Get Pretty Esoteric
If you start hunting around for a simple definition of randomness (maybe that is how you ended up here), you will discover pretty quickly that there is nothing simple about this topic. Different people have different descriptions of randomness depending on the field they work in.
Here, we are largely concerned with statistics, but you can easily get sucked down a wormhole of quantum theory if you are interested in the physics side of things. In +plus magazine, Martin Hairer writes, “For an idea we are all familiar with, randomness is surprisingly hard to formally define. We think of a random process as something that evolves over time but in a way we can’t predict. One example would be the smoke that comes out of your chimney. Although there is no way of exactly predicting the shape of your smoke plume, we can use probability theory – the mathematical language we use to describe randomness – to predict what shapes the plume of smoke is more (or less) likely to take.”
Hairer goes on to explain that randomness appears to be an innate quality of the natural world according to quantum mechanics. He says that if we attempt to send a photon to a beam splitter, it is impossible to predict whether it will pass through or be reflected. One can only maintain that the probability for each is ½. He also explains that there are multiple ways to interpret probabilities, and that even statisticians can get contentious about them.
It is an interesting read, but for our purposes, I did manage to find a simple encapsulation of randomness and probabilities from an academic source. Let’s check it out.
Here is a Simple Definition of Randomness
I found a great definition for randomness here at Carnegie Mellon University. It isn’t clear who wrote it, but it appears to be an upload for a class. It states, “A random outcome is the result of a random phenomenon or procedure.”
“A phenomenon or procedure for generating data is random if
- The outcome is not predictable in advance;
- There is a predictable long-term pattern that can be described by the distribution of the outcomes of very many trials.”
That’s it! Pretty straightforward. This definition also fits with our expectations about casino games.
- We know we cannot predict the next result of a slot machine, roulette wheel, or dice roll;
- But we also know that over a long enough time period, we can expect even distributions of results according to the probabilities listed for each.
Randomness and Probabilities
Returning to Hairer, he says, “The maths for working with probabilities is completely well defined, regardless of your statistical religion.”
That’s reassuring, right? He even gives the following example involving dice:
“For example, suppose I had a six-sided die. Assuming the die is fair, we’d say the probability of rolling any particular number, say a 6, was 1/6. If I wanted to roll an even number, that is, a 2, a 4 or a 6, then I can add the probabilities of these three outcome together as they are mutually exclusive:
- P(even number) = P(2 or 4 or 6) = 1/6 + 1/6 + 1/6 = 1/2 .”
Independent vs. Dependent Trials
Another useful concept to familiarize yourself with when discussing randomness and probabilities is “independent” vs. “dependent” trials or events. Let’s say we have two events. We’ll call them Event X and Event Y.
If the outcome of Event X has an influence on the probability of Event Y, they are dependent events. But if the outcome of Event X does not have any influence on the probability of Event Y,
they are independent events.
Consider dice rolls as an example. Let’s say you roll a die and get a result: 2. Now you roll the same die and get another result: 5.
Getting 2 the first time had zero impact on the second result of 5. These are independent events. There is no causal link. Every number on the first roll had a 1/6 chance of coming up. Every number on the second roll had the same 1/6 chance of coming up.
But now let’s imagine that you want to roll two dice one after the other, and after the first die lands, you want to make a bet about what the two dice will add up to after you roll the second one. So, say the first one lands on 4.
What are the chances that the two dice will add up to 3 after you roll the next one?
You might laugh at this point, because obviously, that is impossible. The count already exceeds 3. But that is exactly the point, and it is a simple demonstration of the fact that we are now talking about dependent events. Your first die has no impact on the result of the second die, but it does impact what the two add up to together. So, each individual roll remains independent, but the outcome of the combined roll is dependent.
Global and Local Randomness
Now that we have talked about independent and dependent events, let’s get into another useful concept to understand: global vs. local randomness.
Even if we grasp intuitively the fact that random events are unpredictable and yield a predictable pattern of distributions in results over numerous trials, those realities are strangely hard to reconcile. We can easily get confused when we are looking at a smaller collection of trials. We know that globally, we should see a nice even distribution, if, for example, we are rolling a die over and over and over again.
If we roll that die millions of times, we should see a distribution like this:
- 1 about 1/6 of the time
- 2 about 1/6 of the time
- 3 about 1/6 of the time
- 4 about 1/6 of the time
- 5 about 1/6 of the time
- 6 about 1/6 of the time
But we intuitively expect to see a similar distribution locally with respect to smaller numbers of trials. If we roll the die six times, in an ideal universe, we think that we should get: 1, 2, 3, 4, 5, 6. We do not expect 2, 2, 2, 2, 2, 2.
Indeed, while placing wagers, we likely would never make such a prediction. And if it happened, we would think, “How improbable!” But the thing is, we need to remember what we discussed about independent events.
If you roll a die six times, none of those trials are dependent on each other. The die itself is agnostic. It does not remember your last roll was a 2. It isn’t “motivated” to roll a 1 or a 3 or a 4, 5, or 6—or a 2, for that matter. So, sometimes you see these long strings of repeating numbers. Or you see other patterns like sequential numbers.
So, how do we end up with a roughly even distribution over millions of trials? The Carnegie Mellon document we referenced earlier for our simple definition of randomness also offers a simple explanation of how we get global expected distributions.
While discussing the presumed “law of averages” and the Gambler’s Fallacy (more on those later), the author says: “This is based on a misunderstanding: that random outcomes achieve regularity by compensating for past imbalances. This is wrong. Instead, random outcomes achieve regularity by swamping out past imbalances.” Hopefully this explanation makes sense to you, because I literally do not think I can explain it better.
Independent trials are not “trying” to balance out past imbalances. If you roll a bunch of 3s in a row, your next die roll is not going to compensate by landing differently. Rather, a pattern of distributions simply emerges given enough repeated trials. The overall global distribution pattern is able to mathematically “swamp out” local imbalances.
So, even if you roll a dozen 3s in a row, over millions of rolls, that unusual event will be just a blip.
Randomness and Hot Cycles, Cold Cycles, and RTP
The discussion above should shed a lot of light on the whole matter of “hot” and “cold” cycles. Let’s consider a few different examples.
Every slot game you play has a statistic called “return to player,” or RTP. This is the percentage of money that players put into the game that we can expect the game to pay out to players over time.
A lot of gamblers do not understand return to player. They think that if a slot machine lists an RTP of 98%, they can expect that 98% of their money will come back to them as they play. They are then astounded and confused at their losing streaks, and wonder, “I play this game a lot, so why is it that I always seem to be losing money? Shouldn’t I only be losing about 2% of my funds?”
Here is why even if you are playing a slot game with a high RTP, you can still lose money:
- RTP is a global quality.
- Spins themselves are independent events.
- Jackpots are outlier events, especially with high volatility slots.
It describes what happens over millions of spins, not what happens over a local series of spins like yours. Just because the RTP for a slot game over millions of spins is 98%, that doesn’t mean that the RTP over a hundred spins or a thousand spins is always that high, or even often.
The slot game is agnostic. The events are truly random, and that means that strings of losses are possible.
If a player wins millions of dollars on a slot game, that is one giant contribution to the RTP. So, the game will pay out less frequently to other players, and smaller amounts.
So, offsetting the occasional big winner are a lot of losers. We can also tie this to the concept of hot and cold cycles. And yes, they exist. As explained by a Luxor slot manager, “Sure they get hot, they also get cold. Through the cycle of a machine it’s percentage to pay out a certain amount over a period of time based on the number of handle pulls the machine receives. However, the hot and cold cycles are random and indeterminable.
Some people say that hot and cold cycles are myths. While they do exist, the actual myth is that you can figure out whether you are “in” a hot or cold cycle. We can define hot and cold cycles like this:
- A hot cycle is simply a period of time when a slot is paying out more frequently than usual.
- A cold cycle is simply a period of time when a slot is paying out less frequently than usual.
But both of these can only be identified in hindsight. You can look back at your previous spins and make a determination about what took place during those spins, but that does not allow you to make predictions about your future spins. The reason you cannot make those predictions is because the slot spins are all independent random events.
Hot and cold cycles are entirely coincidental. There is no causal relationship between the more frequent or less frequent wins of either. Nothing special is taking place behind the scenes. Your next spin has no greater or lesser likelihood of paying out than your previous spin, no matter how “hot” or “cold” the game has been up to that point. So, while the cycles do occur, you cannot use them to your advantage in any way.
Previously, we were talking about RTP, so let’s get back into that in the context of hot and cold cycles as well.
Before, you might have found it weird that hot and cold cycles are even a thing, considering that RTP is a set percentage. But now you understand RTP as a global percentage, and hot and cold cycles as limited local phenomena. Because of the independent nature of the spins and the random nature of the outcomes, they do happen, but they get swamped out over millions of spins, and you end up with the expected RTP.
Roulette is another game of chance where you can experience hot and cold cycles.
Over a very large number of spins, you know the roulette wheel should settle on red or black in an even distribution. Nonetheless, there are times when it lands on red way more than you expect, or black way more than you expect. During those times, either red or black may seem hot or cold. But as with the slot machine spins, the roulette wheel spins produce random, independent results.
Black or red are equally likely outcomes for every future spin, no matter what happened on the previous one, so, you never can predict if the hot cycle or cold cycle you have been experiencing will continue or end with your next spin. Neither possibility is more likely than the other.
When you play keno, you select random numbers as if you were playing the lottery. A computer does the same, and then you see if you picked any matching numbers or not.
There are many different “methods” that people use to try and pick numbers for keno, but as you might guess, some of them are founded in incorrect beliefs about randomness and probabilities.
Right now, if you want, you can look up a list of numbers that are supposedly “hot” for keno right now. You can do this with “cold” keno numbers too. The people who publish such lists may or may not be honest about the fact that they do not really work but there are loads of keno players who buy into them anyway.
Some bet on “hot” numbers to try and leverage the supposed hot cycle for those numbers. Others bet on “cold” numbers to try and leverage the “balancing” they expect the game to eventually have to do to maintain an even distribution.
But now you understand both why “hot” and “cold” number cycles may occur with keno, and also why you cannot use those cycles to get an edge. Over a global number of trials (i.e. millions of keno games), the distribution of numbers selected by keno games should be even. But over a smaller number of trials, imbalances occur, and some numbers show up more than others. Nevertheless, due to the independent nature of the selections, the likelihood of each number being selected by the computer never changes on any round. So, if you see a weird selection like:
80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80
How likely is “80” to be the computer’s next number?
Just as likely as it was before the sequence started.
80 has the same probability of selection on the next pick as 1, 20, 46, or any other number within the range that the keno game includes.
Randomness, the Law of Averages, and the Gambler’s Fallacy
Now that we have talked about hot and cold cycles and RTP, let’s talk a bit about randomness and the Law of Averages as well as the Gambler’s Fallacy.
- What is the Law of Averages?
You will often hear gamblers refer to the so-called “Law of Averages” to justify their bets. The reason that I am putting it in quote marks is because calling it a “law” is not entirely
accurate. Indeed, a more suitable name for it would probably be something more like the “Assumption of Averages.” Context determines the correctness or incorrectness of the assumption.
There is not one definition for the Law of Averages. There are a number of definitions, and they are not identical.
Cambridge defines the Law of Averages like this:
The idea that over a period of time a particular thing will happen because it is just as likely to happen as the other possible events.
The definition Merriam Webster offers is:
The commonsense observation that probability influences everyday life so that over the long term the possible outcomes of a repeated event occur with specific frequencies.
Both of these definitions seem pretty precise, neither of them is particularly clear.
Lumen Learning offers a simpler one:
The law of averages is a lay term used to express a belief that outcomes of a random event will “even out” within a small sample.
This definition is pretty much interchangeable with the Gambler’s Fallacy, which I will address momentarily.
Wikipedia offers another definition for the Law of Averages:
The law of averages is the commonly held belief that a particular outcome or event will, over certain periods of time, occur at a frequency that is similar to its probability.
So, say you have been tossing a coin. You expect that heads and tails will each occur with a 50% frequency over numerous time period, because that is what the probabilities say.
Now, imagine that you get a sequence like the following:
Heads, tails, heads, heads, heads, tails, heads, tails, heads, heads, heads, heads…
There are a disproportionate number of coin tosses that have landed on heads, but the probabilities tell you that 50% of your coin tosses should be tails. So, according to the Law of Averages, you should bet on “tails,” because after all, that will “even things out.” As Lumen Learning points out, the small sample is part of the Law of Averages assumption when it is used in a way that is inaccurate.
Over a very large sample, things will even out, but as we have talked about, it isn’t because of any sort of balancing or compensating. And small samples can be subject to outlier strings.
Merriam Webster actually seems to allude to this since it references the “long term.” That suggests that sometimes when people cite the Law of Averages, they may actually be referencing a truth about probabilities.
- Correct use: “According to the Law of Averages, if I toss this coin a few million times, the outcomes should approach 50/50 for heads and tails.”
- Incorrect use: “I’ve gotten heads fives times in a row. So, according to the Law of Averages, my next toss is more likely to be tails.”
- Dubious use: “I’ve gotten heads fives times in a row. So, according to the Law of Averages, eventually I must get tails.”
Hypothetically, that last one is true. But how long will “eventually” be? There is no telling. I think anyone who bets with a progressive staking system like Martingale has the Law of Averages in mind, and is erroneously assuming they can hang in there until “eventually” happens, but since they are dealing with a small sample size, their losing streak can go on longer
than they expect. And with the rate of increase of their bet sizes, it isn’t long before they wipe themselves out.
What is the Gambler’s Fallacy?
As I mentioned, the Gambler’s Fallacy is pretty much the same as the Lumen Learning definition for the Law of Averages
The Gambler’s Fallacy states:
An event that has been taking place with decreased frequency in the past will take place with increased frequency in the future. Likewise, it also states: An event that has been taking place with increased frequency in the past will take place with decreased frequency in the future. This is exactly the same as saying that one expects things to “balance out” or “even out.”
The Gambler’s Fallacy is the result of believing the Law of Averages applies with a small sample size the same way it does with a very large sample size. It can also result from failing to understand the difference between independent and dependent events.
- The gambler who bets on black after seeing red come up multiple times with roulette is committing the Gambler’s Fallacy if he believes he is making a “smart” bet.
- The gambler who has been losing at slots for the past fifteen minutes who keeps taking spins because things should “surely turn around soon” is committing the Gambler’s Fallacy.
- The gambler who researches a list of “cold” keno numbers and bets on them is committing the Gambler’s Fallacy if he thinks they are “likely” to turn warm, and so forth.
One is not committing the fallacy if one knows better. For example, if our roulette gambler bets on black after multiple reds but does so knowing that black is no more likely than red, he is not subject to the Gambler’s Fallacy, he is simply placing an arbitrary bet and has no unrealistic expectations.
Random Number Generators: How Do They Work?
One more thing you may be wondering about with respect to randomness is how the random number generators (RNGs) that casino games use work. How do they generate truly random results? A variety of different types of random number generators exist.
A lot of RNGs make use of outside data to create their random results. For example, random.org uses atmospheric noise to produce its random results. So, if you use it to, say, come up with a bunch of keno numbers for you to enter in (because you feel like randomizing your inputs), now you know where it got the numbers from.
The RNGs that casinos use do not make use of atmospheric data or other outside data sources. Indeed, they are actually classified as “pseudo-random number generators.” In place of external data to generate their results, they use a seed and an algorithm.
But this should not worry you. Why? Because of the following reasons:
- Hacking this type of RNG would be stupendously challenging. So, it is not likely that anyone would compromise a casino game you were playing and cheat you, nor can you expect to do so yourself to cheat the casino.
- The casino has no motivation to cheat you through their RNG. How so? The house always wins. It is built into the edge with every game. You might think a slot game returning 98% of player funds over time to gamblers would not provide a sufficient profit margin, but that 2% really adds up. Plus, why would the casino want to risk its reputation and its player base by scamming?
- Third parties verify casino RNGs are performing appropriately. You can check out the verifications each casino you play at has received from independent testers, and you can research the reputations of the testing agencies. If there were a problem with the RNG of any game that a casino was offering, the agency would find out, and the casino would need to fix the problem immediately.
- Casinos would lose their licenses if it was discovered that their games were not producing genuinely random outcomes. This is not in any serious business’s best interest.
We have discussed the concepts of randomness and probability from a number of angles now, and how they connect to the Law of Averages, the Gambler’s Fallacy, RTP, hot and cold cycles, dependent vs. independent events, and large vs. small sample sizes. We have also talked a bit about how random number generators work. Let’s review our key takeaways:
- The outcome of any individual trial of a random event is not predictable, but a distribution pattern arises over many, many trials according to the probabilities.
- Events in gambling may be independent or dependent. If they are independent, there is no causal link to connect them, and their outcomes are not causally related either.
- Global randomness results in expected distribution patterns (i.e. 50/50 heads and tails for coin tosses). Global randomness requires many, many trials.
- Local randomness is subject to more outlier events (i.e. repeated heads or tails).
- Global randomness does not involve events “compensating” or “balancing” for past imbalances in order to result in the expected distribution. Instead, that distribution arises naturally as imbalances get “swamped out.”
- RTP is a global measurement. It involves millions of spins.
- Hot and cold cycles do exist, but we can only describe the past as hot or cold. We cannot make predictions about the future based on that past. The probabilities for independent events are constants. No matter how many times a coin has landed on heads, “heads” still has the same 50% chance of coming up next as it ever did.
- The “Law of Averages” is a confusing concept with varying definitions. We can say accurately enough that it is a law that with numerous trials, one should see events occur with a particular expected frequency. But it is incorrect to assert that the “Law of Averages” applies with a small number of trials.
- When gamblers make the mistake above with the Law of Averages, they are committing the Gambler’s Fallacy.
- Random number generators can either be truly random, deriving their data from outside sources (like atmospheric noise), or pseudo-random, using a seed and an algorithm.
- Casinos use pseudo-random number generators, but the results they produce are random and fair, as verified by third-parties.
Best Practices While Gambling With Respect to Randomness
Given everything we have gone over, what should you do with respect to randomness while playing at online casinos?
Well, the whole thing about randomness is there really is not much to do about it. Acknowledging randomness is acknowledging our own limitations as human beings, but here are some general recommendations:
- Always, always, always keep in mind everything you learned in this article. That way you will not commit the error that is the Gambler’s Fallacy. You do not want to lose money because you do not understand randomness.
- Do choose slots with high RTPs, but understand that doing so is only saving you money if you gamble a lot over a very long time. It is not going to have a noticeable impact on your bottom line in the near future.
- Think carefully about whether you are dealing with independent or dependent events before you make bets. In dice games, for instance, you can encounter both.
- Avoid progressive staking systems if you want to hang onto your bankroll. You cannot rely on the Law of Averages to bail you out as you stake ever higher amounts of cash on loss after loss.
- Do not go on tilt trying to “win it all back” at a slot machine or other random game. If you do this, you are subconsciously buying into the Gambler’s Fallacy and/or incorrectly applying the Law of Averages.
- Know the odds with any event you are wagering on, and manage your bankroll accordingly.
- Do not take “hot” and “cold” cycles too seriously. Yes, they happen, and you will experience them. But since you cannot predict whether a hot or cold cycle will continue, they should be a source of neither hope nor despair. Literally anything could happen on your next spin.
- If you do feel your emotions getting out of hand, take a break.
Regardless of the games you play and the approaches you ultimately end up taking, randomness and probabilities are fascinating, and there is no better way to wrap your head around them than to play online casino games.
So, have fun discovering more about what randomness is and how it works. Maybe you’ll get lucky along the way and win big!
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