Jan 02 2012
One concept that is important to being a scientist or critical thinker is that terms need to be defined precisely and unambiguously. Words are ideas and ideas can be sharp or fuzzy – fuzzy ideas lead to fuzzy thinking. An obstacle to using language precisely is that words often have multiple definitions, and many words that have a specific technical definition also have a colloquial use that is different than the technical use, or at least not as precise.
Recently on the SGU we talked about randomness, a concept that can use more exploration than we had time for on the show. The term “random” has a colloquial use – it is often used to mean a non-sequitur, or something that is out of context. It is also used colloquially in the mathematical sense, as a sequence or arrangement that does not have any apparent or actual pattern. However, people have a generally poor naive sense about what is random, mathematically speaking.
There are at least two specific technical definitions of the term random I want to discuss. The first is mathematical randomness. Here there is a specific operational definition; a random sequence of numbers is one in which every digit has a statistically equal chance of occurring at any position. That’s pretty straightforward. This operation can be applied to many sequences to see if they conform to a statistically random sequence. Gambling is one such application. The sequence of numbers that come up at the roulette table, for example, should be mathematically random. No one number should come up more often than any other (over a sufficiently large sample size), and there should be no pattern to the sequence. Every number should have an equal chance of appearing at any time. Otherwise players would be able to take advantage of the non-randomness to increased their odds of winning.
Computer simulations are another area where a truly random sequence of numbers is valuable. Random numbers may provide the input necessary for the simulation to run.
It is very difficult (perhaps impossible) for a person to generate a truly random sequence of numbers from their brain. Here are three sequences of numbers, try to find the one that is mathematically random:
0 4 4 7 2 0 6 0 2 3 8 9 9 3 0 2 0 5 3 3 8 6 8 4 9 3 3 8 9 2 4 2 2 1 3 6 4 7 9 7 4 0 2 4 9 9 3 4 5 0
9 4 8 5 7 6 0 9 4 7 3 6 5 2 9 1 7 3 5 7 8 5 4 8 0 2 9 3 8 7 5 1 0 2 5 2 3 5 5 5 0 2 9 8 9 7 7 2 0 3
8 5 5 7 0 3 0 9 2 9 9 2 8 4 7 5 6 6 2 0 3 9 4 8 7 5 0 3 0 9 4 8 7 5 0 3 0 3 8 4 7 5 9 8 7 7 0 3 9 8
The top sequence was generated by a random number generator, the bottom two I produced by typing chaotically (I won’t say “randomly”) on my number pad. The top sequence is statistically random, while the bottom two are not. It’s hard to tell the difference just by looking. Also we tend to underestimate the clumpiness of randomness (called the clustering effect). So, for example, in a mathematically random sequence of numbers, the same digit should occur twice in a row with a certain frequency, and even three, four, five or more times in a row. But such clusters make the sequence look naively non-random.
The top number is what is called pseudo-random. As I said, random numbers are very useful to computer programmers. There are a number of operations that can generate mathematically random number sequences. But they are not truly random because the operation will generate the same sequence of numbers given the same input or seed. There therefore needs to be some way to create a random seed, which can be based upon some physically noisy process, or the time, or something else that changes regularly.
Another example of a pseudo-random sequence is pi. The number pi (3.1415926535897932384626433832795028841971693993751058209 7494459230781640628620899862803482534211706798214808651…) is a statistically random sequence of digits, but of course it is not truly random because it is one specific sequence.
This brings us to the second technical definition of random – true physical randomness. I can throw dice to generate a statistically random sequence of numbers, assuming the dice are fair and I am sufficiently “randomizing” each throw. But from a physical point of view, the result of each throw is not random, but determined by the laws of physics. The number that results on the die must occur given all the physical parameters of the throw. Once the die is cast, the number that will result is determined and not random. In this sense “random” also means “unpredictable.”
The only truly random physical system known to science results from quantum effects. Certain quantum properties are undetermined and unpredictable – they are truly random. In fact, researchers last year developed a random number generator based upon quantum properties – the first truly random number generator.
As with many concepts in science and elsewhere, even seemingly basic or simple concepts can become very detailed and complex when explored deeply. That is one lesson I have thoroughly learned from studying and teaching science – it’s always more complicated than it seems. In fact, it’s always more complicated than your current understanding. The above discussion of randomness is a quick overview, but there are layers of complexity and detail I did not get into. There are also limits to our current understanding – the universe is more complicated than we know.
It is very helpful, however, to at least understand that there is likely more depth to an issue than one’s current knowledge. But we can still use terms and concepts that are accurate and precise as far as they go, even if there is always a deeper complexity.
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