Why is the roll of a die considered random?

I've been reading articles on pseudo-randomness in computing when generating a random value. They all state that the generated numbers are pseudo-random because we know all the factors that influence the outcome, and that the roll of a die is considered truly random. But I'm wondering why. Don't we know all the physical forces that influence the die when it's being rolled? Or is there too many of them?

Solution 1:

A die roll is only considered random if the external factors are not controlled.

Practiced dice cheats can roll numbers they want to roll. So talk about nerves and blood vessels and quantum effects are just wrong. These cheats control the meaningful factors such that they influence the outcome of the roll, predictably. Even if someone only increases their chance of rolling a certain number by a few percentage points, that's huge in gambling terms.

That's why there are rules on how the dice must be rolled at casinos, and inventions such as the dice tower:

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Solution 2:

This has to do with chaos theory: the tiniest variation of the initial conditions will cause an enormously different output. For a physical system like a die toss:

• even from a classical point of view, it is very unlikely that you can know the very exact initial conditions of the throw. And of the environment: the "floor" distance and surface characteristics (think of the abrupt effect of each bounce, that will be very different depending on the most infinitesimal variation of the impact parameters), the air conditions (thermodynamic and kinematic)...!

• this becomes actually impossible if you include the uncertainty principle (that prevents you from knowing the exact value of certain pairs of variables at the same time, e.g. position and momentum, but see below);

• it would be impossible from a practical point of view to propagate these initial conditions without introducing round-off errors, that due to the chaotic nature of the problem would make the result completely unreliable;

• even if you could perform exact calculations, there is still the quantum indeterminacy (again, see below) that affects the development of the status of the die: at each bounce, even when air molecules brake the die rotation, it is impossible even theoretically to predict what will happen in the next instant with absolute certainty.

As pointed out in many comments and with many downvotes, the contributions to the randomness of the roll from quantum effects are insignificant from any practical point of view. Nevertheless I do want to mention them since they provide a theoretical watertight border against a deterministic idea of the phenomenon.

Taking care of another possible correct objection, I have to underline that my answer holds for a fair throw. If you think of a die "tossed" from, say, $1\,\mathrm{mm}$ above a horizontal flat floor, with negligible initial velocity and a face parallel to the ground, it is obvious that you can predict the outcome with practical certainty. Moving progressively away from this limit situation, you have many halfway toss styles that can influence the probability distribution of the outcomes, if only by a few percent. I'm referring to the opposite limit, when the system can be considered ergodic. When I heard this term applied to the die, maybe not $100\%$ properly, it was with the meaning that the system "scans" over time all the possible outcomes many many times, with equal probability and with no recognizable pattern. Add the fact that a fair throw starts with a random grip of the die, and you really have equal chances for all the outcomes.

Solution 3:

The roll of a die is modelled as being random.

The purpose of a mathematical model is to help us to understand some feature of the world.

A real die falling onto a surface is a mechanical system. It can be modelled by a deterministic mechanical model, which could be used to generate pseudorandom numbers. However, this is not usually a particularly useful model. Instead we can use a uniform discrete distribution as a model for the die. This is a much better model. It simplifies the mechanical system, and allows you to gain insight into the world. It allows you to consider the expected value and variance of the die roll (these concepts would be hidden by the detail of the mechanical model).

In mathematical questions phrases like "Yusuf rolls a fair die..." are part of the code: It means "There is a random variable with a discrete uniform distribution on {1...6}. The question is asking you to form a mathematical model. A more sophisticated model takes into account the slight biases due to the mass of the die not being centred, or the sides not being identical.

This is what is meant by a die being random: It is an object that is usefully modelled by a random variable.

Solution 4:

I feel the existing answers all concentrate on the dice and would like to offer another perspective, concerned with the generator:

The "pseudo" in "pseudo random generator" means that the next value is completely determined. There is no surprise at all, no matter what you do. The only "surprise factor" arises if you do not happen to know the internal state of the generator. But it is trivially easy to get the internal state (just read out the RAM), and it is there.

Also, if you restore a known internal state, you can repeat the series of "random" values you got before. This is actually, where PRNGs are used, often a welcome feature - for example, you can generate a lot of, well, pseudorandom data from a very small seed value. So to communicate whatever you generated, you only need to transfer the seed, not all values. Or you can test something which needs seemingly random input in a repeatable way (for debugging, demonstration etc.).

For dice, or many other physical processes, the problem is not so much that we are too stupid to read out the state, but that there actually is no deterministic process underlying whatever we are using to generate the next random value. It is not so much about us not being able to figure it out ("read the RAM") but that there "is no RAM". For dice, the previous roll of the dice has no relation whatsoever to the next roll. Sure, we might, in ideal conditions and with stunning advances in measuring apparatuses, somehow try to predict the current roll while it is happening. But even if we'd manage to do that (quantum effects, Heisenberg, chaotic effects ignored), this roll would have no relation whatsoever with the next roll. Hence we call it truly (independently) random.

If you are more interested, a nice read is "The Art of Computer Programming" by Knuth. He spends a lot of time on PRNGs, including how to measure how "random" a random generator actually is.

Solution 5:

Even without invoking Quantum Theory, some systems are chaotic. In principle, they can be predicted but this requires impossibly perfect measurements of the initial state. Quantum theory prohibits these perfect measurements but such precision would be required that even without quantum mechanics, sufficiently perfect measurements would not be possible.

One nice example is a toy which is a consists of a pendulum which contains a magnet over a base with two magnets; the magnets are aligned to attract the pendulum. Place the base magnets slightly either side of the rest point of the pendulum. Move the pendulum off centre and release it. It will come to rest over one of the magnets but which? There will be areas around each base magnet in which the pendulum will predictably stop over that magnet. However, outside those areas, the behaviour is very complex. Here is one of the first hits that I found in a search, I expect that there are better ones: A pendulum and two magnets - an example of a chaotic system.

A purer mathematical example is the Mandelbrot set. In principle it is utterly predictable. Any point is either in the set or not. However, in some areas, this cannot be determined in finite number of steps. Mandelbrot set at Wikipedia

Another example is the solar system. Assume that Newton was right and ignore everything except the Sun and major planets (include Pluto if you wish). With a perfect measurement of the initial state and perfect calculations, we could predict its future indefinitely into the future. However, the tiniest error could render the results seriously wrong later on.