Thursday, September 23, 2010

Conjunction and Probability

What do we do in ordinary life when we have to calculate the probability of finding a sequence repeated, where the events that are conjoined are not known to be connected in some other way?

Suppose we throw a die and get a six. The chances that six will turn up in a single throw are one in six, or 1/6. Assuming that we know of nothing in the way the die is thrown that will secure one side rather than some other, the chances that a second throw will have the same result is 1/6 x 1/6 or 1/36. The chances of getting three sixes in a row is (1/6) * (1/6) * (1/6), or 1/216. The probability of getting six sixes in a row would be one in almost 50,000, and if we carried the repetitions to ten or more, the figures would become astronomical.

If we found that anyone kept getting sixes regularly, we would begin to get suspicious. We would accept their results perhaps through two or three repetitions, or perhaps a few more. But there would come a time when the hypothesis of mere luck would strain our belief, compared to the theory of some sort of control, that we would only a dupe would continue believing it.

We can't say what the chances are of getting any one result, because we can't set a limit to the number of possible alternatives (Strictly speaking, we can't set such a limit in the other case either. We can't rule out beforehand the possibility that the die would stand on one of its corners and spin there permanently.). But we can say something if one of these results goes on repeating itself to the exclusion of all other results.

If there is nothing in one event that would compel or require another event to follow, then it's more likely that the first event would be followed by by something other than some other third event than by the second event itself. And if, in spite of this, only the second event continues to appear with the first event, it's just as naive to assume they are connected by anything but chance, as it would be to assume the same thing about winning throws of the die.

But that's what the denial of intrinsic relations commits us to. It says that if water continues to put out fires, it's just luck.

But if you believe in merely chance conjunctions, then among the possibilites covered by that view is not only repetition of the first event with varying consequents, but also its repetition with the same result.

It's always the improbable that happens. Watever side of a die comes up, the chances agsint it before it happend were five to one. Yet for all that, it happened. The simultaneous occurrence of the events composing the present universe is only one out of an infinite number of possibilities, and hence was all but infinitely improbably a moment ago. Yet here it is.

In the same way, the repetition of two events toegther a dozen times, or a hundred, or a thousand, is unlikely beforehand, but is still one of the possibilities conceivable under the rule of chance. Therefore, it's absurd to offer it as evidence against the rule of chance.

I don't think there is a reply to this, on it's own grounds. I think there are other ways in which the theory that all succession is chance can be refuted, such as by demonstrating necessity in inference. Out of an infinite number of chance combinations, the known history of the world may present one, and it might be argued that any extension of that history, with any multiplication of its uniformities, could only produce another.

But the argument commits the fallacy of inexhaustive division. The question at issue is how the successions we find are to be interpreted, and the original argument proceeded by offering the alternative of chance or some form of necessity, and then eliminating chance. The alternatives considered in the reply are not these. These are combinations that would be possible only by chance.

You've been so eager to show that your hypothesis of chance will cover the facts that you've forgotten that to the hypothesis under which all your combinations fall, there is a further alternative which may cover the facts equally well. And when we turn from chance to intrinsic connection, it not only covers the facts much better.

For on the chance hypothesis, every successive repetition of a conjunction given in the past is the occurrence of the progressively more improbable, while with intrinsic connection, it's only a confirmation, more impressive at each recurrence, of what the hypothesis predicted. So far, neither theory can exclude its rival. But the chance hypothesis, while consistent with the known arrangement, would accord equally with any other arrangement. The hypothesis of connection accords selectively with the arrangement actually found.


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