27 August 2005

What is Randomness?

image of pair of dice
Luck, Chance, and Randomness -- Three ideas that most people don't have a clue about.

If a sequence is unpredictable, we call it "random". Stated another way, outcomes which lack discernible patterns are said to be "random". We don't define randomness by what it is, but by what it isn't. Of course an apparently "random" sequence may have an underlying order, we just don't (yet) know how to find it.

"Common sense" makes it hard for us to grasp the importance of randomness when we encounter it. We tend to see patterns in series of numbers that are really "random" -- we see patterns and seek causes when there are no patterns. People think that random events repeat less often than they actually do, so when they see repetitions they assume non-randomness. This causes most people to view some events as being significant or meaningful when they really aren't.

Such patterns include runs of successful outcomes in sports or games of chance. The basic fallacy is that the outcome of past events can influence future events. In random series the past has no effect on future outcomes, by definition.

Why Is Randomness Important?

Internet Security --

The usefulness of the internet depends on security. If you are going to send or receive financial information, or anything else confidential, you have to believe that the various parts of the internet you will use are not easily read by other users.

This involves cryptographic systems to hide your information. These systems depend on random numbers. Deterministic machines like computers cannot, in principal, generate true randomness. This creates an interesting race between programmers trying to create security systems (trying to create keys with as much randomness as possible) and those trying to penetrate those systems (by discovering the non-random elements in the keys).

Here is an interesting article about how two computer scientists broke the encryption system of Netscape Navigator a few versions ago.

Gambling --

Most casinos and other sponsors of gambling depend on the gambler's misconceptions about randomness and probability. Gamblers believe they can predict random systems. Casinos know a lot more about randomness than gamblers do. Sports gambling is not so different, since sports results are much more random than bettors believe.

Some state lotteries use computer-generated numbers, and some algorithms are better than others. (As noted above, no computer can generate a truly random series.)
A Canadian mathematician once deduced the algorithm used in his province's lottery, and he proceeded to win the next two lotteries. The authorities arrested him for fraud. The judge ruled that the mathematician, although clever, was not a criminal and let him keep the money. The next lottery used a different (and harder to figure) algorithm! (This may be just an urban legend -- I can't find any other source for this story.)
In fact lotteries go to considerable lengths to achieve impenetrable randomness. This page describes how the California Lottery numbers are picked. (Note that winning numbers in some California games are picked by "automated draw machines", i.e. computers, and are thus perhaps less random (that is, predictable if you can find the algorithm used) than those using physical systems.)

The published numbers from lotteries actually provide a useful source of random numbers for various purposes.

Pop Quiz:

How many times would you have to shuffle a deck of cards to be sure they were in completely random order? Answer: You can't truly randomize the order of a deck of cards by shuffling. But you can make them pretty random. Shuffling a deck two or three times isn't enough to destroy the order that existed in the deck before shuffling. In fact, repeated perfectly accurate shuffling can return the deck to its original order! To get a deck randomized enough for practical (gambling or card-playing) purposes takes more than seven imperfect shuffles. Here's a good summary of the problem.

"The Gambler's Fallacy" --

If you have flipped a coin five times, and each time it has come up "heads", what do you think will be the result of the next toss? If you thought, "It should come up heads half the time, and tails half the time, so it must be due to come up tails on the next toss," you fell for the gambler's fallacy. You felt that somehow the coin can "remember" past outcomes. It's really no more likely to come up tails on the next toss than to come up heads. That string of five heads in a row would happen 3% of the time (in a really large number of random flips) just by chance.

Science --

Those who analyze big bunches of numbers to try to find patterns (scientists, stock-market investors, sports nuts) use statistics. How can you recognize when a certain pattern of numbers represents some real effect of some real cause, and when it is just the chance result of random fluctuations? Most science depends on identifying departures from randomness.

Experimental research also depends on "randomized" procedures, to prevent the (perhaps unconscious) bias of the experimenters from influencing the results. This creates a big demand for random numbers.

"The Hot Hand Fallacy" --

Patterns seen in a series of outcomes, whether dice throws or winning games, are thought to be due to some unseen cause, rather than the random series that they probably are. The team is said to be "on a streak". This is rarely the case. Remember we saw above that a run of five heads in a row in a coin toss series is to be expected from time to time, even though the outcome of each toss is unpredictable.

On the other hand, this analysis indicates that past success may affect future throws in horseshoes. This pdf page discusses the analysis and implications of apparent "hot" streaks in basketball and other sports.

Are day-to-day fluctuations in stock-market prices random, or is there an underlying pattern? Check out this article on Random Walks.

Quantum Mechanics

"I think it is safe to say that no one understands Quantum Mechanics." (Richard Feynman, 1918-1988)
Quantum processes are thought to be random. Thus the series of radioactive decay events in a radioactive sample would create a random series, with no predictability. This idea has been used to try to create random series for practical uses. This site is one source of such random series. Random.org uses another method, and has a discussion of methods available on line to get "random" numbers.

Additional Resources

Some links to discussions of randomness, especially in computers.

Unusually, Wikipedia is weak in this area. But its article on randomization is interesting.

The Vietnam-era draft lotteries were government efforts to randomize men's chances of being called up. This site shows how non-random the 1970 drawing was.

An interesting post about the search for randomness.

A good site for exploring randomness. Can you behave randomly?

A little old, but an interesting discussion of using networked PCs to crack internet encryption schemes.

A page devoted to the "hot hand" in sports.

Technorati tags: , , , , , ,


Statisfactions said...

New link for the article about horseshoes: http://economics-files.pomona.edu/GarySmith/horseshoes.html

Mark Marciano said...

This was very interesting to read especially the part on sports. So often do we see hot streaks and cold streaks with teams and something that people don't do is look at the actual randomness behind it. Also with the coin flip and the coin not being able to recall prior sequences is something I've never thought about. So often do we bank on a coin not being heads 5 times in a row but in reality the randomness of it could make it happen.