What do you mean “Random”?

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People use the word random pretty casually, but it is actually a big deal. What would it mean, for example, for a crowd in a movie to look “random”? Oddly, it seems to be one-sixth women and five-sixths men — this is a result found by the Geena Davis Foundation, which examines the issues of representation of women in media. Since there are slightly more women than men (women live slightly longer), a really “random” crowd would, logically, be about half women, probably slightly more. Asking audience members when a crowd looks balanced returns a positive response when the crowd is composed of about one-sixth women. When the crowd is one-third women, audience members think that the crowd is majority female. This is a simple, if startling, example of how human beings are terrible at determining if something is random.

People equate “random” with “fair”.

When one of two people must be chosen, a go-to solution is “flip a coin”. Drawing straws, one of which is shorter, is a random process and so it is viewed as being a fair way to pick someone to wash the dishes on a camping trip or clean up the mess the puppy made. In an earlier Occupy Math, there are several examples of probability problems that people get wrong all the time. Consider this: our inability to determine if something is actually random is a potential source of systematic unfairness.

If we believe a group that is one-sixth women is balanced between men and women, then, in some sense, we believe that giving women one-sixth of the slots in a group is fair and natural. This is an additional causative factor for gender discrimination. Can you think of a TV show in which there is a group of four to six characters, one of whom is female? Penny in the first season of Big Bang Theory. Black Widow in Avengers Assemble. This phenomenon is so common it has a name: The Smurfette Principle, which is also a killer example of a group with one woman.

What does it mean to be random?

If we say that something is random if it occurs only by chance without intervention, planning, or human influence, then all we’ve done is to define one type of non-randomness (human interference) and export the problem of figuring out what “random” means to other words. Since Blaise Pascal figured out how to beat the French National Lottery several centuries ago, there has been a real market for numbers that are actually random. Silicon Graphics Corporation holds a US Patent for a proveably random number generator based on image processing of pictures of lava lamps. It is called Lavarand and covered under U.S. Patent 5,732,138, titled “Method for seeding a pseudo-random number generator with a cryptographic hash of a digitization of a chaotic system”. A widely held and probably correct belief is that quantum mechanics is actually random — meaning, for example, that we can use radioactive decay events as a source of true random numbers.

What if we want to characterize randomness by a test? First of all, one number by itself is not random because randomness is a statement about what happens if you sample a process over and over. This means that only a sequence of numbers can be random. An early test was that a sequence of numbers was considered random if each digit occurred (very close to) the same number of times. In 1933, D. G. Champernowne came up with the following idea: list the counting numbers in ascending order. Each digit occurs one-tenth of the time in this sequence and yet this sequence is violently non-random.

But what is a random sequence, then?

It turns out that Champernowne’s notion had a lot of cousins. Randomness test after randomness test was defeated by the construction of a simple and obviously non-random process that generates a sequence of numbers that pass the test. Eventually, the discipline of computer science found a correct — and relatively impractical — test for randomness. A sequence is judged to be non-random if there is a pattern to the numbers. This then leads to the following definition: The degree of randomness of a sequence is the minimum amount of computational effort needed to demonstrate the sequence is not random. A truly random sequence, like the sequence of times between positron emissions of a sample of sodium-22, cannot be shown to have a pattern and so infinite computational effort is insufficient. The definition works, but it is really hard to prove something is random; the definition is much more useful for demonstrating something is non-random. The digits of the constant π, for example, are clearly not random because there is a simple algorithm that types out as many digits of π as you are willing to wait for.

What do people mean by random?

Most people’s intuitive definition of randomness is the situation in which all possibilities are equally likely. This is called uniform randomness. This is also a bit tricky. If you flip a fair coin (heads and tails are equally likely), then the coin is uniformly random, but if you flip a fair coin twenty times then ten heads are much more likely than two heads. This shows that randomness is strongly dependent on viewpoint and it gets us back to the problem with people thinking a crowd has a uniformly random sex distribution when it is one-sixth women.

In statistics there are objective tests (e.g. the Chi-squared test) for determining if a sample is clearly not following a specified distribution, including a uniform one. This means that, if you’re willing to adopt a particular type of randomness, then you can check and see if something that is supposed to be happening according to that type of randomness is bogus. Since most people don’t use statistics, this means that unfairness and discrimination arising from incorrect perceptions of randomness are going to be a persistent problem.

Occupy Math hopes this discourse on the importance and opacity of randomness is useful, if only by highlighting the complexity and subtlety of the term “random”. The sequence of ones and zeros at the top of this post is not random at all. Can you find the rule for generating it? Occupy Math promises there is a simple one. Can you think of other places where our inability to correctly sense randomness is a problem? Please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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