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of belief upon ignorance." It defines rational expectation by measuring the comparative amounts of knowledge and ignorance, and teaches us to regulate our actions with regard to future events in a way which will, in the long run, lead to the least disappointment. It is, as Laplace happily said, good sense reduced to calculation. This theory appears to me the noblest creation of intellect, and it passes my conception how two such men as Auguste Comte and J. S. Mill could be found depreciating it and vainly questioning its validity. To eulogise the theory ought to be 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 case resembling it in the necessary circumstances. The theory consists in putting similar cases on a par, and distributing equally among them whatever knowledge we possess. Throw a penny into the air, and consider what we know with regard to its way of falling. We know that it will certainly fall upon a side, so that either 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 also 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; it would consist in treating unequally things of which our knowledge is equal.

The theory does not 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 the 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 reason for expecting head more than tail. Our state of knowledge will be changed should we throw up the coin many times and register the results. 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 side in the long run; trial might show that the die was a loaded one, and falls 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.

To

It may be asked, as Mill asks, Why spend so much trouble in calculating from imperfect data, when a 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. 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 divergent results 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, 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 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. It is the comparative numbers of ways in which events can happen which measure their comparative probabilities. 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. 184) 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 similarly draw all the following probabilities:One combination gives o tail. Probability. Three combinations gives I tail. Probability Three combinations give 2 tails. Probability One combination gives 3 tails. Probability 1.

We can apply the same considerations to the imaginary causes of the difference of stature, the combinations of which were shown in p. 188. There are altogether 128 ways in which seven causes can be present or absent. 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 2. The probability of five feet three inches is; of five feet one inch 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.

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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, as far as is known, are 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. 178), 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

We must distinguish comparative from absolute probabilities. In drawing a card casually from a pack, there is no reason to expect any one card more than any other. Now, there are four kings and four queens in a pack, so that there are just as many ways of drawing one as the other, and the probabilities are equal. But there are thirteen diamonds, so that the probability of a king is to that of a diamond as four to thirteen. Thus the probabilities of each are proportional to their respective numbers of ways of happening. Again, I can draw a king in four ways, and not draw one in forty-eight, so that the probabilities are in this proportion, or, as is commonly said, the odds against drawing a king are forty-eight to four. The odds are seven to seventeen in favour, or seventeen to seven against the letters R,o,m,a, accidentally forming a significant word. The odds are five to three against two tails appearing in three throws of a penny. Conversely, when the odds of an event are given, and the probability is required, take the odds in favour of the event for numerator, and the sum of the odds for denominator.

It is obvious that an event is certain when all the combinations of causes which can take place produce that event. If we represent the probability of such event

according to our rule, it gives the ratio of some number to itself, or unity. An event is certain not to happen when no possible combination of causes gives the event, and the ratio by the same rule becomes that of o to some number. Hence it follows that in the theory of probability certainty is expressed by I, and impossibility by o; but no mystical meaning should be attached to these symbols, as they merely express the fact that all or no possible combinations give the event.

By a compound event, we mean an event which may be decomposed into two or more simpler events. Thus the firing of a gun may be decomposed into pulling the trigger, the fall of the hammer, the explosion of the cap, &c. In this example the simple events are not independent, because if the trigger is pulled, the other events will under proper conditions necessarily follow, and their probabilities are therefore the same as that of the first event. Events are independent when the happening of one does not render the other either more or less probable than before. Thus the death of a person is neither more nor less probable because the planet Mars happens to be visible. When the component events are independent, a simple rule can be given for calculating the probability of the compound event, thus-Multiply together the fractions expressing the probabilities of the independent component events.

The probability of throwing tail twice with a penny is xor; the probability of throwing it three times running isxx, or ; a result agreeing with that obtained in an apparently different manner (p. 202). In fact, when we multiply together the denominators, we get the whole number of ways of happening of the compound event, and when we multiply the numerators, we get the number of ways favourable to the required event.

Probabilities may be added to or subtracted from each other under the important condition that the events in question are exclusive of each other, so that not more than one of them can happen. It might be argued that, since the probability of throwing head at the first trial is, and at the second trial also, the probability of throwing it in the first two throws is, or certainty. Not only is this result evidently absurd, but a repetition of the process