well said that the quantity of belief is always relative to a particular state of knowledge or ignorance; but it must be observed that it is absolute in the sense of not being relative to any individual mind; since, the same information being presupposed, all minds ought to distribute their belief in the same way.' Dr. Boole, too, seemed to entertain a like view, when he described the theory as engaged with the equal distribution of ignorance,' but we may just as well say that it is engaged with the equal distribution of knowledge. I prefer to dispense altogether with this obscure word belief, and to say that the theory of probability deals with quantity of knowledge, an expression of which a precise explanation and measure can presently be given. An event is only probable when our knowledge of it is diluted with ignorance, and exact calculation is needed to discriminate how much we do and do not know. The theory has been described by some as professing to evolve knowledge out of ignorance; but as Professor Donkin has admirably remarked, it is really a method of avoiding the erection of belief upon ignorance.' It defines rational expectation by measuring the comparative amounts of knowledge and ignorance, and teaches us to regulate our action with regard to future events in a way which will, in the long run, lead to the least amount of disappointment and injury. It is, as Laplace as happily expressed it, good sense reduced to calculation. This theory appears to me the noblest creation of human intellect, and it passes my conception how two men possessing such high intelligence as Auguste Comte and J. S. Mill, could have been found depreciating it, or even vainly attempting to question its validity. To C 'Philosophical Magazine,' 4th Series, vol. i. p. 355. d Transactions of the Royal Society of Edinburgh,' vol. xxi. part iv. e Philosophical Magazine,' 4th Series, vol i. p. 355 eulogise the theory is as needless as to eulogise reason itself. Fundamental Principles of the Theory. The calculation of probabilities is really founded, as I conceive, upon the principle of reasoning set forth in preceding chapters. We must treat equals equally, and what we know of one case may be affirmed of every other case resembling it in the necessary circumstances. The theory consists in putting similar cases upon a par, and distributing equally among them whatever knowledge we may possess. Throw a penny into the air, and consider what we know with regard to its mode of falling. We know that it will certainly fall upon a flat side, so that either the head or tail will be uppermost, but as to whether it will be head or tail, our knowledge is equally divided. Whatever we know concerning head, we know as much concerning tail, so that we have no reason for expecting one more than the other. The least predominance of belief to either side would be irrational, as it would consist in treating unequally things of which our knowledge is equal. The theory does not in the least require, as some writers have erroneously supposed, that we should first ascertain by experiment the equal facility of the events we are considering. So far as we can examine and measure the causes in operation, events are removed out of the sphere of probability. The theory comes into play where ignorance begins, and the knowledge we possess requires to be distributed over many cases. Nor does the theory show that the coin will fall as often on one side as the other. It is almost impossible that this should happen, because some inequality in the form of the coin, or some uniform manner in throwing it up, is almost sure to occasion a slight preponderance in one direction. But as we do not previously know in which way a preponderance will exist, we have no more reason for expecting head than tail. Our state of knowledge will be changed, indeed, should we throw up the coin many times in succession and register the result. Every throw gives us some slight information as to the probable tendency of the coin, and in subsequent calculations we must take this into account. In other cases experience might show that we had been entirely mistaken; we might expect that a die would fall as often on each of the six sides as on each other one in the long run; trial might show that the die was a loaded one, and fell much the most often on a particular face. The theory would not have misled us: it treated correctly the information we had, which is all that any theory can do. It may be asked, Why spend so much trouble in calculating from imperfect data, when a very little trouble would enable us to render a conclusion certain by actual trial? Why calculate the probability of a measurement being correct, when we can try whether it is correct? But I shall fully point out in later parts of this work that in measurement we never can attain perfect coincidence. Two measurements of the same base line in a survey may show a difference of some inches, and there may be no means of knowing which is the better result. A third measurement would probably agree with neither. To select any one of the measurements, would imply that we knew it to be the most nearly correct one, which we do not. In this state of ignorance, the only guide is the theory of probability, which proves that in the long run the mean of different quantities will come most nearly to the truth. In all other scientific operations whatsoever, perfect knowledge is impossible, and when we have exhausted all our instrumental means in the attainment of truth, one there is a margin of error which can only be safely treated by the principles of probability. The method which we employ in the theory consists in calculating the number of all the cases or events concerning which our knowledge is equal. If we have even the slightest reason for suspecting that one event is more likely to occur than another, we should take this knowledge into account. This being done, we must determine the whole number of events which are, so far as we know, equally likely. Thus, if we have no reason for supposing that a penny will fall more often one way than another, there are two cases, head and tail, equally likely. But if from trial or otherwise we know, or think we know, that of 100 throws 55 will give tail, then the probability is measured by the ratio of 55 to 100. The mathematical formulæ of the theory are exactly the same as those of the theory of combinations. In this latter theory, we determine in how many ways events may be joined together, and we now proceed to use this knowledge in calculating the number of ways in which a certain event may come about, and thus defining its probability. If we throw three pennies into the air, what is the probability that two of them will fall tail uppermost? This amounts to asking in how many possible ways can we select two tails out of three, compared with the whole number of ways in which the coins can be placed. Now, the fourth line of the Arithmetical Triangle (p. 208) gives us the answer. The whole number of ways in which we can select or leave three things is eight, and the possible combinations of two things at a time is three; hence the probability of two tails is the ratio of three to eight. From the numbers in the triangle we may draw all the following probabilities : Probability One combination gives o tail. Three combinations give 2 tails. Probability 3. One combination gives 3 tails. Probability. We could apply the same considerations to the imaginary causes of the difference of stature, the combinations of which were shown in p. 213. There are altogether 128 ways in which seven causes can be combined together. Now, twenty-one of these combinations give an addition of two inches, so that the probability of a person under the circumstances being five feet two inches is. The probability of five feet three inches is 2; of five feet one inch is; of five feet, and so on. Thus the eighth line of the Arithmetical Triangle gives all the probabilities arising out of the combinations of seven causes or things. 21 Rules for the Calculation of Probabilities. I will now explain as simply as possible the rules for calculating probabilities. The principal rule is as follows: Calculate the number of events which may happen independently of each other, and which are as far as is known equally probable. Make this number the denominator of a fraction, and take for the numerator the number of such events as imply or constitute the happening of the event, whose probability is required. Thus, if the letters of the word Roma be thrown down casually in a row, what is the probability that they will form a significant Latin word? The possible arrangements of four letters are 4 × 3 × 2 × 1, or 24 in number (p. 201), and if all the arrangements be examined, seven of these will be found to have meaning, namely Roma, ramo, oram, mora, maro, armo, and amor. Hence the probability of a significant result is 7 f 24. f Wallis 'Of Combinations,' p. 117. |