# Why is the term “Monte Carlo simulation” used instead of “Random simulation”? [duplicate]

I always read/hear "Monte Carlo" simulations. I have done "Monte Carlo" simulations before to calculate the odds in certain gambling games as part of my job and it was nothing more than basically using an RNG to simulate random results (slot machines spinning wheels) and build upon them to get the final result of the game, repeat and get an estimated average outcome.
Anyone would be doing this without ever knowing that what they're doing is called "Monte Carlo" X.

My question is, is there any reason to call a random simulation a "Monte Carlo" simulation other than sounding sophisticated/clever? Any technical/legitimate reasons?

EDIT: I suggest people read this question's answers. It represents what I wanted to ask about better. I was trying to find out if there are technical reasons to distinguish between a "Monte Carlo" and a "random" simulation, disregarding any historical reasons.

• see history here: en.wikipedia.org/wiki/Monte_Carlo_method#History – Aksakal Dec 27 '17 at 20:31
• My guess is that using "Monte Carlo" gives the informed readers that there was some statistical underneath the motivations of the speaker? "Random Simulation" can mean a lot of things to statistician if not clearly defined. – dylanjm Dec 27 '17 at 20:53
• As a side note: in randomized algorithms, there exist the two classes of Monte Carlo and Las Vegas algorithms, so there the name serves as a distinction. (@Anyone, feel free to incorporate this into an answer if you are so inclined) – Oliphaunt - reinstate Monica Dec 27 '17 at 23:35
• A random sumulation is imprecise. I can imagine several things one would describe as "random simulation". But the "Monte Carlo" method is exactly one method which is sharply defined. So precision is the answer. Its the name for an established method. If you want to refer to that method and want to be easily understood, refer to it using the established term, don't make your own terms up which have the potential for confusion. Replace "random simulation" with "Frobnication method" and then make your case why it should be called that. If you can't, then stick to Monte Carlo. – Polygnome Dec 28 '17 at 14:31
• In light of your edit, isn't the question you reference now an exact duplicate? How does your question differ from it? – whuber Dec 28 '17 at 15:51

## 5 Answers

Basically, all Monte Carlo methods use the (weak) law of large numbers: The mean converges to its expectation.

Then there are Quasi Monte Carlo methods. These are simulated with a compromise of random numbers and equally spaced grids to yield faster convergece.

Simulations that are not Monte Carlo are e.g. used in computational fluid dynamics. It is easy to model fluid dynamics on a "micro scale" of single portions of the fluid. These portions have an initial speed, pressure and size and are affected by forces from the neighbouring portions or by solid bodies. Simulations compute the whole behaviour of the fluid by calculating all the portions and their interaction. Doing this efficiently makes this a science. No random numbers are needed there.

In meteorology or climate research, things are done similarly. But now, the initial values are not exactly known: You only have the meteorological data at some points where they have been measured. A lot of data has to be guessed.

Further, Monte Carlo Simulations are expected to help researchers obtain results close to reality, they are random simulations meant to mimic reality. If your random simulation doesn't have anything to do with reality or predicting some actual event, then it would not be correct to call your random simulation a Monte Carlo Simulation.

• IMHO your answer, Jack Meister's answer and the the answers in the question you like, like the one by @Tim, are the best so far – SpaceMonkey Dec 28 '17 at 15:44

It was at that time that I suggested an obvious name for the statistical method - a suggestion not unrelated to the fact that Stan[islaw Ulam] had an uncle who would borrow money from relatives because he “just had to go to Monte Carlo.”

"Monte Carlo" refers to the eponymous casino in Monaco. Of course, as you note, casinos have a connection to random number generation. (And to - potentially ruinous - results from generating many random numbers.)

This nomenclature needs to be seen in the context of a group of physicists and mathematicians that amuse themselves playing small-stakes poker. Relatedly, Stanislaw Ulam wrote in his memoirs that

Metropolis once described what a triumph it was to win ten dollars from John von Neumann, author of a famous treatise on game theory. He then bought his book for five dollars and pasted the other five inside the cover as a symbol of his victory.

This may give you an idea of the intellectual environment that gives birth to technical terms patterned after places of gambling.

Edit: you ask

is there any reason to call a random simulation a "Monte Carlo" simulation other than sounding sophisticated/clever?

I don't see or know of any other reason other than it's the commonly accepted term for a random simulation. This may not be a "technical" reason, but I would say that using an accepted term for a technical issue is quite a sufficient reason to minimize misunderstandings.

• I wasn't asking about the historical reason why it's called Monte Carlo really, but thanks anyway – SpaceMonkey Dec 28 '17 at 0:00
• @Spacemonkey What's in a name? That which we call Monte Carlo, By any other name would be as random. – uhoh Dec 28 '17 at 4:12
• Then I misunderstood, sorry. I'll go with @Nij's comment: the reason to use an established term instead of some other one is that people understand what you are talking about. Far easier than to try to establish one's own nomenclature. – Stephan Kolassa Dec 28 '17 at 8:15
• @AdamDavis the fact that it is a well understood term is legitimate reason. The fact that it is the term least likely to create confusion can be also understood as technical reason. It is official name of that technique, and that's it. It's like asking if calling you Adam has any real reason other than sounding biblical, as if the fact that it is your name doesn't matter. – Mołot Dec 28 '17 at 11:22
• Space, your objection is puzzling. Historical reasons are perfectly good explanations for terminology (as well as often being interesting, authoritative, and convincing). After all, your stated question is "is there any reason to call a random simulation a "Monte Carlo" simulation." – whuber Dec 28 '17 at 14:48

I have sometimes heard of people who distinguish between Monte Carlo algorithms and Las Vegas algorithms. Unlike a Monte Carlo algorithm—which will always terminate, but has a chance of giving wildly inaccurate results—a Las Vegas algorithm has a chance of running for an arbitrarily long time, but will always give accurate results. I suspect that most people don't make the distinction often, since (as you note) most people use "Monte Carlo" and "random" interchangeably. (Wikipedia says that there are also Atlantic City algorithms, but I had never heard of that term until now.)

• This doesn't address the term Monte Carlo. Like Las Vegas and Atlantic City, Monte Carlo is a famous gambling location and since gambling involves chance and random number generation is done by chance that was the connection. It is also called simulation. – Michael R. Chernick Dec 28 '17 at 5:16
• The issue of termination indicates you aren't writing about the statistical meaning of a Monte Carlo simulation at all: you have a particular kind of (nondeterministic) algorithm in mind. Although they share a name and randomness in their implementation, they are distinct ideas with different applications. – whuber Dec 28 '17 at 14:43
• @whuber: I see what you mean. Consider a "fuzzy sort" algorithm that just runs randomized quicksort for 10 steps. This would be a Monte Carlo algorithm by my definition, but it wouldn't be a Monte Carlo simulation (as the question asks about) because there is no model being simulated. – Jack Meister Dec 28 '17 at 21:22

You can read about the history of the Monte Carlo name in the other answers and comments. So this answer will provide a complementary perspective.

In sophisticated company, it's referred to as stochastic simulation. See for example, the book "Stochastic Simulation: Algorithms and Analysis", Asmussen and Glynn. http://www.springer.com/us/book/9780387306797 .

Monte Carlo simulation is a rather down-market term (pardon my snobbery). In my workplace, I usually refer to Monte Carlo simulation, because many people wouldn't have a clue what I was talking about if I said stochastic simulation. I don't usually find myself in upscale company there, ha ha.

• I wonder what kind of people you'd like to hang out with, if you don't consider the likes of Neumann, Metropolis or Ulam "sophisticated". Also great to hear you consider your coworkers so highly :) In any case, I don't think your answer adds anything to the question - the question is about "why Monte Carlo instead of just random?", and you basically just said "Actually, some people call it Stochastic Simulation." The interesting thing is why X instead of random simulation, not all the other ways to call the same thing. – Luaan Dec 27 '17 at 23:29
• @Luaan the other answer by Stephan doesn't answer my question either. I wasn't asking about how the term was coined, I was asking if there is any real reason to use this term other than that other people use it. – SpaceMonkey Dec 28 '17 at 0:02
• @Space monkey Could it have been called random simulation instead of stochastic simulation? Yes. But it evolved (after Von Neuman, Metropolis) from being called Monte Carlo simulation to being called (in certain academic circles) stochastic simulation. Similarly, stochastic processes could instead be called random processes, but they are often called stochastic processes. Apparently, my partially tongue-in-cheek response was too subtle for some people to appreciate, The "sophisticated" company is certain academic circles, but outside them, the hoi polloi rule, and that's where I usually am. – Mark L. Stone Dec 28 '17 at 0:50
• The only reason for using any label is to ensure other people know exactly what you're talking about. Many people are familiar with calling it Monte Carlo simulation, and since random simulation is an ambiguous phrase (is the randomness in how you conduct your simulation, how you chose the simulation, how results are obtained, or what?) the other option is clear without being overly technical. The origin of the name is the reason to use that name, as opposed to Atlantic simulation or my backyard simulation or any other arbitrary choice. @Spacemonkey – Nij Dec 28 '17 at 0:55

This is actually a really good question, and it has provoked some fine answers. I'm adding this because I wondered about this too, and believe that Monte Carlo was used and became popular, because the process employed by gambling casinos, and the statistical process of estimation have specific, similar characteristics. The methods employ randomness, but themselves are not random, as scientists typically use the term. (Eg. "I expected to see some evidence of xxxxxx, but the results looks completely random."). Both the Monte Carlo casino operators, and those using statistical techniques are seeking a specific outcome, and they are using similar methods to achieve a desired outcome. Random typically implies no evident pattern or unpredictable.

The methods used by gambling casinos are very well thought out. The unpredictability of the specific outcome of specific events is established (else it would not be gambling, would it?), but the nature and distribution of the range of outcomes is fully understood - and this key fact, both in casino gambling, and in the use of statistical estimation techniques - makes all the difference.

An example: A Monte Carlo roulette wheel will have 1 to 18 numbers in one colour, and 19 to 36 in another, if I remember correctly, so red or black have equal probability of appearing. You can wager a specific number, or just bet on red or black appearing. The players are playing against each other, for each other's money. How does the casino operator make any money from this?

The wheel has a "0" position, and if and when the ball lands there, the casino operator rakes in all the bets - the house wins. Each time the ball lands on the zero, the house wins. So each trial - each spin of the wheel - has a random outcome, but the house has (assuming the wheel is not rigged, say by putting a little magnet under the "0" number), then the house still can expect to - on average - sweep all the bets off the table with a 1/37 (or 0.027027) probability. And if the house wants to improve its outcome? It can add a second number to the wheel - typically "00", or double-zero. Now, the probability that the house will win is almost (but not quite) doubled, to 2/38 (or 0.0526316). That's over 5%, or a serious take. Suppose the average amount of money wagered at the table each night, by the high-rollers, is $170,000. With one zero, the house can be expected to make 170000 * 0.027027 =$4594.59, but adding the extra zero, and the expected take for the house is now 170000 * 0.0526316 = \$8947.37.

See, the amount gambled each night will be random. We won't know what it will be. But assuming the wheel is fair and true (and smart gamblers are always watching to see if a game is "rigged" - just like the house detectives are always watching to see if players are cheating), we can say with close to certainty, that by adding the extra zero to the wheel, the casino can almost double it's take from the roulette game. If the casino operators can add that extra zero to the wheel, and not drive away players, and reduce the nightly amount bet, then they will do it. And just using simple probability, the improvement in the cash-take can be predicted. If the casino operator adds free drinks, to attract more players, then the cost of such extra attractions can be subtracted from the expected improved take. It may well be that adding the extra double-zero to the wheel and adding free alcohol, may improve the bet-flow. As the casino operator, you would run some experiments, and assess the outcomes. And since the operation of the process is filling your pocket with money, you are willing to focus on how it works, with some serious attention to detail.

And that, lastly is the key point. Although randomness is employed, the operation of any casino is a very analysis-intensive business, where statistical techniques and an understanding of probability are key to obtaining a successful outcome. Casino operators know the expected outcome on each and every game they offer, and because the randomness is limited to activity within a known distribution of possible results, the expected cash take on each game can be estimated quite accurately. This is how cheaters are caught. One particular game experiences a big divergence from expected outcome, right? As the casino operator, you know something is wrong.

Monte Carlo methods, or the techniques of statistical and probabilistic estimation, can be very effective at predicting the outcomes of processes where the distribution of possible results is known. And if either side can obtain an "edge", or shift the distribution of random outcomes in such a way as to alter the long-term expected value of the outcome even slightly, such an "edge" can make a person (or more often, the casino operator), very rich.

Monte Carlo methods employ randomness, but the methods - and the outcomes they can provide - are not random at all. The methods themselves can be as well-engineered and finely-tuned as the engines of the Porsche automobiles in the casino parking lots. And that is why the term is used.

• just adding something here: "effective at predicting the outcomes of processes where the distribution of possible results is known", that is not true because if the distribution was already known, you wouldn't need an estimation of the outcome, you would calculate it exactly (which is what we did + use simulation to verify the model). The simulation gives you the distribution as well as the the expected outcomes (with a confidence interval), all you need is the mechanics of the game. – SpaceMonkey Dec 28 '17 at 17:41
• Inputs are known but as Space Monkey notes the outcomes are not. Sometimes Monte Carlo has been used to "confirm" conjectures. – Michael R. Chernick Dec 28 '17 at 18:51