happen, or positively that it will not. But its probability to us means the degree of expectation of its occurrence, which we are warranted in entertaining by our present evidence.
Bearing this in mind, I think it must be admitted, that even when we have no knowledge whatever to guide our expectations, except the knowledge that what happens must be some one of a certain number of possibilities, we may still reasonably judge, that one supposition is more probable to us than another supposition; and if we have any interest at stake, we shall best provide for it by acting conformably to that judgment.
§ 2. Suppose that we are required to take a ball from a box, of which we only know that it contains balls both black and white, and none of any other color. We know that the ball we select will be either a black or a white ball; but we have no ground for expecting black rather than white, or white rather than black. In that case, if we are obliged to make a choice, and to stake something on one or the other supposition, it will, as a question of prudence, be perfectly indifferent which; and we shall act precisely as we should have acted if we had known beforehand that the box contained an equal number of black and white balls. But though our conduct would be the same, it would not be founded on any surmise that the balls were in fact thus equally divided; for we might, on the contrary, know by authentic information that the box contained ninety-nine balls of one color, and only one of the other; still, if we are not told which color has only one, and which has ninety-nine, the drawing of a white and of a black ball will be equally probable to us. We shall have no reason for staking any thing on the one event rather than on the other; the option between the two will be a matter of indifference; in other words, it will be an even chance.
But let it now be supposed that instead of two there are three colors—white, black, and red; and that we are entirely ignorant of the proportion in which they are mingled. We should then have no reason for expecting one more than another, and if obliged to bet, should venture our stake on red, white, or black with equal indifference. But should we be indifferent whether we betted for or against some one color, as, for instance, white? Surely not. From the very fact that black and red are each of them separately equally probable to us with white, the two together must be twice as probable. We should in this case expect not white rather than white, and so much rather that we would lay two to one upon it. It is true, there might, for aught we knew, be more white balls than black and red together; and if so, our bet would, if we knew more, be seen to be a disadvantageous one. But so also, for aught we knew, might there be more red balls than black and white, or more black balls than white and red, and in such case [pg 381] the effect of additional knowledge would be to prove to us that our bet was more advantageous than we had supposed it to be. There is in the existing state of our knowledge a rational probability of two to one against white; a probability fit to be made a basis of conduct. No reasonable person would lay an even wager in favor of white against black and red; though against black alone or red alone he might do so without imprudence.
The common theory, therefore, of the calculation of chances, appears to be tenable. Even when we know nothing except the number of the possible and mutually excluding contingencies, and are entirely ignorant of their comparative frequency, we may have grounds, and grounds numerically appreciable, for acting on one supposition rather than on another; and this is the meaning of Probability.
§ 3. The principle, however, on which the reasoning proceeds, is sufficiently evident. It is the obvious one that when the cases which exist are shared among several kinds, it is impossible that each of those kinds should be a majority of the whole: on the contrary, there must be a majority against each kind, except one at