Probability

Equally Likely Possibilities.

Why is it stipulated that the alternative possibilities must be equally likely? The point can be made clear by comparing two simple cases. Suppose one were to argue that a properly balanced coin when tossed must either throw head or not, so that there are two alternatives, of which head is one, and therefore its probability must be 1/2. The answer would be true, but the reasoning would be wrong, as may be seen from the next case. Suppose one were to argue that a properly con structed die, when thrown, must either show face six or not, so that there are two possibilities of which face six is one, and therefore its probability is 1/2. Here the answer is obviously wrong. But why? Because the possibility "not-six," that is, of face six not showing, really represents five different possibilities against the one "six," and must therefore be weighed accordingly as having a value five times that of "six." The probability of "not six" is really 5/6, while that of "six" is 1/6. It was a mere acci dent that in the case of the coin "not-head" happened to represent only one alternative, namely, "tail." If no attention were paid to the equality of the alternative possibilities, then any statement, the truth of which we are not in a position to judge, would be judged to be as likely to be true as not—some logicians have actually assigned it a truth-probability of //2 It would be just as accurate to say that there is a probability of 1/2 that an un known marksman will hit the bull's eye—an absurd estimate when it is remembered that there are innumerable places, on and off the target, which are all included under "not bull's eye" as against the very limited area of "bull's eye." If the available data are insufficient, why estimate the probability at all? Why not suspend judgment? There is no particular virtue in emulating Bagehot's village maiden who must have an opinion about every thing under the sun.

The next question is, how is one to make sure whether the alternative possibilities really are equally likely, or at least approximately so? Sometimes this is not very difficult to deter mine. In the case of a coin, or a die, e.g., it is not impracticable to examine them sufficiently closely and ascertain whether or no they are properly balanced. And if properly balanced, the possi bilities which they offer would be adjudged to be equally likely.

But suppose one cannot be sure. Is there not some other way of testing the equality of the possibilities? Well, there is—namely, by actually tossing the coin, or throwing the die, a great many times, and noting the, results. If, on an average, each side of the coin, or of the die, appears approximately an equal number of times, then the alternative possibilities would be regarded as equally likely. But the significance of this kind of test must be noted carefully, if only because of its bearing on the inductive, or a posteriori calculation of probability.

This kind of test is not direct but indirect, or inverse, in the sense in which induction is said to be inverse deduction. The logic of the test is this. We argue that if the possibilities repre sented, say, by the several faces of the die, are really equally likely, then when the die is cast a sufficiently large number of times, each face should, on an average, and in the long run, appear approximately once in six throws. This rate of appearance is called its frequency. So that this mode of procedure may be described as consisting in testing the equality of the possibilities by reference to the frequency which seems to be implied in the probability calculated on the assumption of their equality.

Now, in calculations of probability, "frequency" and "proba bility" are so frequently identified or confused that it is important to mark their difference. When the a priori probability, say, of face six appearing when a die is cast, is given as i/6, what this really means is that "the appearance of face six is one of six equally likely possibilities"; and so always f/t simply has refer ence to possibilities, which may never be tested at all. One never thinks of adding the expression "in the long run" to the fraction expressing a probability. On the other hand, when a frequency is stated, even or especially when it is expressed by the same fraction as the corresponding probability, the phrase "in the long run" is essential, otherwise the frequency given would be wrong, in view of the series of long runs of some one number, which are so common.

We may now turn to the a posteriori calculations of probability, a method the true character of which has already been suggested in the preceding remarks.