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The theory of probability uses random variables, which avoids the need to define what one single random choice means.

Yet in everyday conversations about probability, even professional probabilists often talk about making one single random choice.

For definiteness, suppose we are mainly interested in the question of making one single random choice from the half-open unit interval [0, 1) with the uniform distribution.

Does there exist a logically consistent, rigorous notion of a single random choice? Why or why not?

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  • Randomness is one of those concepts which resists definition. We are not even sure if randomness exists. People often use the term as a synonym for unpredictability, but we know that randomness is not the same as unpredictability. Certain mathematical theories define randomness - algorithmic information theory says that a sequence is random if it cannot be compressed.
    – nwr
    Commented Apr 18, 2022 at 18:29
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    @nwr For practical purposes, randomness is the same as unpredictability. Then apply the pragmatic maxim: "Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object."
    – causative
    Commented Apr 18, 2022 at 18:56
  • causative, I don't know about "randomness is the same as unpredictability". At least in the theory of probability, a random process is assumed to be unpredictable in a systematic way, as defined by a fixed assignment of probabilities to a certain family of events (subsets of the sample space). This is known as a "probability law". But certainly there can be unpredictable events that follow no law. Commented Apr 18, 2022 at 22:47
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    Not an answer, but I'll just drop it here: plato.stanford.edu/entries/chance-randomness ... lots and lots of reading for quiet winter evenings.
    – AnoE
    Commented Apr 19, 2022 at 8:27
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    What do you mean by random choice? Are you asking how I define that I don't know the result of a coin flip, or are you asking how I would generate an unpredictable real number in the range [0,1), or are you asking about a more rigorous definition of a random variable in the said range?
    – Mankka
    Commented Apr 19, 2022 at 9:24

3 Answers 3

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It's best to think of randomness as a model, not as a 'thing'. When we talk about a 'random event', we mean that at some time t0 we cannot predict with certainty the state of a system at time t1 (t1 > t0) because of the influence of some intervening event. We may be able to quantify our uncertainty — that's the business of statistics — but we cannot eliminate it.

We have, so to speak, reached a proverbial fork in the road, and have no way of calculating precisely which path will be taken.

Keep in mind that identifying the existence of a 'random event' of this sort doesn't tell us anything much about the event itself. It might be:

  • A truly random phenomenon, such that even a perfect knowledge of the prior state of the universe would not let us predict the next instant.
  • An unknown (or at least unmeasured) force that we could analyze if we wanted to
  • A non-linear (chaotic) deterministic process that could in theory be predicted if we knew the entire exact state prior to the instant, but where short of that, better data does not generally allow better predictions.
  • A known and expected force that is merely too complex to calculate with ease, and is dealt with more easily as an error factor
  • An extrinsic influence such as human will or the act of some god

We do not know, and perhaps cannot know, so we model it as 'random' and work with it as best we can.

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  • Reminded me of Yogi Berra's aphorism on randomness: "When you come to a fork in the road, you should take it". Commented Apr 20, 2022 at 10:31
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I want to try and give an answer to this question from a "objective Bayesian" perspective. Objective Bayesianism is the idea that probability measures should be thought of as corresponding to states of knowledge that an observer might have. The "objective" part refers to the idea that in some situations there is a "correct" state of belief that one should have, and therefore a correct prior to use.

For an objective Bayesian, to say something is uniformly distributed on [0,1) means that they don't know its numerical value (beyond it being at least 0 and less than 1). The probabilities do not represent randomness in this case, but just how likely the person thinks the value is to fall into a given range. This can be formalised in terms of the odds they would accept on a bet, or in several other ways.

So far there isn't any notion of "random choice", or randomness at all - the person could simply be reasoning about an unknown fixed value. But suppose we want to construct something, physically, such that our state of knowledge about it is the uniform distribution. That means we need to create a value, somehow, such that (i) we don't immediately know it, and (ii) according to the knowledge we do have, it is as likely to be within one sub-interval of [0,1) as another of the same length.

It's kind of hard to think about how you'd do that for a real number (and we can get into all sorts of gnarly issues about how accurately it's possible to measure something), so let's switch to a more familiar example: we want to choose a number that's uniformly distributed on the set {1,2,3,4,5,6}.

That means we want to create something such that (i) it has a definite integer value between 1 and 6, (ii) we don't immediately know what that value is, and (iii) any value between 1 and 6 is as likely as any other.

The obvious solution to this is to roll a die. Assume for now that we roll it behind a screen, so we don't know what value it has. Then we have a value between 1 and 6 that we don't know, and if we assume the die is properly made and rolled well, then we don't have any reason to think any value is more likely than any other.

We don't have to roll the die behind a screen either - we can just say that before we roll the die there is still a fact of the matter about what face will come up, it's just that we don't know what that is until we see it. After all, when you roll a die you're just imparting some momentum to it. The face that comes up depends on that initial momentum in a very complicated way, depending on collisions with the table and with air molecules and so on, but in principle it could be calculated - it's just very hard to do so.

(One can argue about whether that's true or not - are quantum effects important in the rolling of a die, and if so are they "truly random" in a way that classical mechanics isn't? What about the free will of the person rolling the die? - but I will neglect these arguments because they're not really crucial to the point. If you prefer, instead of a die you can think of a random number generator on a computer, where none of those arguments apply.)

For a real number between [0,1) you might do something like spin a disk with the numbers from 0 to 0.9999 marked on the circumference. Or you could roll a 10-sided die a large number of times to get the decimal expansion, e.g. 0.1853724... (Both of these have the issue that you can't really get an infinite number of decimal places, although the second one allows you to get as many as you need for any given purpose.)

In short, I'm proposing that "a single random choice" could be defined as something like "a process designed to produce a result that is unknown for a given observer, and such that for that observer their subjective probability distribution has a given form."

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  • A previous version of this answer referred to subjective Bayesianism instead of objective Bayesianism, but on reflection I think this is closer to an objective Bayesian view. This is because we're trying to set things up to make a particular prior the "correct" one. In general this answer feels like something Edwin Jaynes would have agreed with, who was probably the most prominent member of the objective Bayesian school.
    – N. Virgo
    Commented Apr 26, 2022 at 0:50
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Philosophically speaking random choice is an oxymoron. Random chance is the very opposite of deliberate choice. Both mean the selection of one option out of many. The difference is that a choice is always intentional, made for a purpose, while chance is unintentional, serving no purpose.

Randomness in general is widely misunderstood concept, because it has several different meanings, which may confuse many.

In mathematics randomness is a property of a series, the lack of pattern, the unpredictability of the next value based on previous values.

In physics randomness is the probabilistic inaccuracy in all events. Causes don't determine their effects with absolute accuracy, all effects are partially random.

In philosophy and common parlance randomness means lack of intent. We meet random (unchosen) people, we draw random (unchosen) cards from a deck, we throw dice to get random (unchosen) results.

It would make things easier if we had different words for different kinds of randomness.

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