It's probability time again, folks! Pull up a comfy chair, and I'll tell you a tale of Really Big Numbers.

Today's story is prompted by the dear people over at the Internet Infidels Discussion Forums. A participant going by the name of "elwoodblues" posted a poll asking people to vote on whether they thought a really large number of monkeys typing for vast amounts of time would eventually produce a page of Hamlet. At the time I looked, fewer than forty people had voted on it, so each vote is still making a fair difference to the outcome, but the initial results show around two thirds of people thinking that this was a reasonable proposition. The author followed it up with a brief mathematical analysis, then a little later realised that I'd beaten him to the punch (in 1995) with the venerable The Mathematics of Monkeys and Shakespeare.

But hey! What are the chances of someone else producing something so similar to my nutty old essay like that? Well, I'm glad I asked, because that's an interesting question, and if you know the answer to it (and other questions like it), you can annoy people by being a smart-alec. So, for all you aspiring smart-alecs out there, pay attention to the master as I now reveal that the answer is "exactly one". Why "exactly one", you ask? Because it actually happened. The nature of the question was historical: it asked about an event that we'd already seen, and thus there was no element of randomness involved. When you ask about a concrete past event rather than a potential or theoretical event, the "probabilities" are always exactly one (meaning that it happened) or exactly zero (meaning that it didn't). So there.

Be warned, however, that making this smart-alec remark runs the usual risk of receiving a slap upside of the head in retort. The person asking the question usually means to ask "what were the chances of that?" — an entirely different question. But even if they are clever enough to phrase it in a way that denies you the obvious opportunity for recreational pedantry, you can still engage in some smart-alec nitpicking. In order to compute the probability of something, you see, you need to specify the desired outcome or set of outcomes, and the mathematical details of how outcomes are reached. If you want to phrase the probability in terms of a time-frame, then you also need to know how often outcomes are reached.

Let's run through a trivial example to demonstrate this. Suppose I roll a common gaming die, and it comes up "six". "Wow!" exclaims a convenient hypothetical victim, "what were the chances of that?" I leap upon the opportunity to explain to my convenient hypothetical victim, in excruciating detail, what a complex question this is. Hidden in the innocent phrase, "the chances of that", lie a multitude of assumptions. These assumptions include such important factors as the number of faces on the die, the number that each face shows, the assumption that the die is unbiased towards any particular face when rolled in the usual way, and the assumption that its properties do not change from roll to roll. Most importantly, however, we assume the "that" in his question equates to "the number six, particularly, being topmost on any given roll of the die".

This last assumption is of paramount importance, and it may not be as sound an assumption as it appears at first glance. Why not? Because, as we are about to discover, this convenient hypothetical victim is easily impressed. After explaining that the chances were "one in six" (contingent on all the assumptions I mentioned earlier), I roll the die again and it comes up "two". Again, my victim is impressed, and asks what the chances were of that happening. If I'm not careful here, I'm going to become the convenient victim, because the chances of this person being impressed by the outcome of any particular die roll is "one". No matter what number comes up, he's going to be impressed that it only had a one-in-six chance of doing so, yet it beat the odds and came up anyhow. In one of life's ironic twists, my victim is incapable of realising that he's being impressed by something happening when there was no chance of it not happening. The rules of the game mean that some number is going to come out on top no matter how many times I roll the die.

Imagine how impressed my victim would have been if I'd been rolling a hundred-sided die. Each number that came up would be beating hundred-to-one odds! At a time like that, I'd wish I had access to a perfect sphere. I would tell my victim that it was an infinite-sided die, and that the chances of anything coming out on top were therefore impossible. "Wow! Let's roll it and see what happens!"

Speaking of hundred-sided dice, a poster using the handle "Jesus Tap-Dancin' Christ" produced a comment in the Infidels' discussion forum which makes exactly the kind of error I'm lampooning here. He claims that he can generate events of arbitrarily large improbability on demand using his large collection of N-sided dice. The trick is, of course, that he's just going to roll one die after another, and let it come up "something". The chances of a die coming up "something" are "exactly one" for any theoretical die (and likewise for any practical die that doesn't shatter on impact or meet some other such inconvenience that real-world dice encounter from time to time). Running these events independently, one after the other, means that their probabilities (of "exactly one") multiply to produce a final probability of, well, "exactly one". It comes as no surprise to anyone that "Jesus Tap-Dancin' Christ" can produce events of probability "one" on demand. It takes divine intervention or a good dose of cheating to produce any other kind of probabilistic event on demand, but the only way not to produce a probability "one" event is not to try!

Let's be clear about this. If you can produce an outcome without fail, on demand, then it is, by definition, a probability "one" outcome. I don't care whether it involves dice or not: if you can make it happen first time, every time, then it's probability "one". If you want to experiment meaningfully with probability, then roll your dice in such a way that some outcomes are considered "successful", and some are considered "failures". Call a "one" success with everything else a failure, or vice versa: you'll get a much better understanding of what odds mean that way, looking at your ratio of "successes" to "failures", and the number of consecutive successes or failures you have.

Don't fall for the idea that extremely improbable things happen all around you all the time, or at least not in any way that makes probability irrelevant. If we are really talking about probability, then something with a one-in-N chance of happening happens, on average, once in every N attempts. The chances of winning the first division prize in one of the lotteries in my state are slightly worse than seven million to one against, but people win it quite frequently because so many tickets are sold. Nobody ever wins that lottery three times in a row, however, because the chances against it are 351,754,079,835,298,748,608 to one against. If six billion people played the lottery every day of the year for over a hundred and sixty million years, you could reasonably expect it to have happened once, but we aren't living in anything like that kind of universe, so we quite reasonably expect it not to happen at all.