We therefore say that the chance of getting that face-say the six-is. Since the ace and the six each occur in onesixth of the total number of throws, one or other of them will occur in one-third of the total number; and so we say that the chance of getting either an ace or a six is, and so on. It will be noticed in this case that the throw which gives any one of the six faces cannot by any possibility give any other; and so we can make some such general statement as this: Where two or more events are incompatible the chance of getting either one or the other is found by adding together the fractions which express the chances of each. Of course the chances against any given event or alternative are found by subtracting the fraction in favor of it from 1.

When we are tossing two dice (A and B) instead of one, we expect that in the long run each of the following combinations will occur about as often as any other:

With reference to the 6 or any other given face we can

summarize these results as follows:

Putting these results into more general form:"If the

I

chances of a thing being p and q are respectively and

m

I

n

I

then the chance of its being both p and q is

mn

9 is

(m

1)(n

1)

mn

where p and q are independent. The sum

of these chances is obviously unity; as it ought to be, since one or other of the four alternatives must necessarily exist.

Four more

One thing that a non-mathematician is liable to overlook in these figures is this, that the throws in which we get a six with either of two dice are not so common as the throws in which we get either a six or an ace with one cautions. die. We turn up as many sixes with the two dice as we turn up sixes and aces with one; but since the two sixes are on different dice and are therefore not incompatible, they come together in one throw out of thirtysix, and we do not turn them up in so many separate throws. This explains the necessity for the word 'incompatible' in the formula which we gave on page 339.

A second thing to notice about the table has been already referred to in another connection: namely, that if we add together the numbers on the two dice in each throw, we shall find that one sum is by no means as common as another. Seven is the commonest, for it can be made by six different combinations; 6 and 8 next; then 5 and 9, and so on until we reach 2 and 12, each of which occurs only once. once more the mean is commoner than the extremes.

Thus

A third thing about these tables is worth dwelling upon because we are all likely to forget it when the figures are not before us: namely, the extremely small number of cases in which two independent improbable events coincide. Sixes with a single die are thrown in one case out of six, but double sixes with two dice in only one out of thirty-six, and if we should guess double sixes as often as they are thrown (ie., one time out of thirty-six) the guess would be right

* Venn, p. 174.

(ie., coincide with the throw) in only one case out of 36 × 36, i.e., in one out of 1296. To take another example of the same thing, if the chances of taking a certain disease are, and if a first attack neither increases nor decreases the liability to a second, the chance of a given person having that disease twice is only TOOOO· In other words, if one person out of 100 has it once, only one out of 10,000 will have it twice. The difference between these two figures is of course very striking, and any one who sees them is likely to forget about the mathematics and jump to the conclusion that a first attack affords almost complete immunity against a second. As Wallace says in his very ingenious (though by no means conclusive) article against Vaccination: "Very few people have smallpox a second time.' No doubt. But very few people suffer from any special accident twice— a shipwreck, or railway or coach accident, or a house on fire; yet one of these accidents does not confer immunity against its happening a second time. The taking it for granted that second attacks of smallpox, or of any other zymotic disease, are of that degree of rarity as to prove some immunity or protection, indicates the incapacity of the medical mind for dealing with what is a purely statistical and mathematical question."* Unfortunately" the medical mind" is not the only one that is likely to forget how rapidly fractions diminish when they are squared.

The method of ascertaining causal relations by comparing the number of actual coincidences between two events or circumstances with the number that would naturally be produced by mere chance according to the theory of probability is being used more and more as statistics of various sorts become more and more available; and by this method we must expect to reach many conclusions that seem at first, for the reason just given, to be contrary to all experience.

A fourth word of warning about the interpretation of

* Alfred R. Wallace, "The Wonderful Century ", N. Y., 1898.

tables of probability. That an event turns out so many times in a given way is no reason why we should act that many times as though we expected it to turn out that way. On the contrary we should act each time as though we expected it to turn out in the most probable way. If we are guessing the total number of spots turned up by two dice we will be right in 6 cases out

But if we should guess each

and if we guess 7 every time, of 36, or 216 out of 1296. number as many times as that number is actually turned up, we should be right in only 146 cases out of 1296, as the following table shows :

Total number of correct guesses 146.

What is true in this case is true in any other: we should act each time as though we expected the most probable outcome to be found then. The figures show how much we can afford in the long run to risk upon each guess. They do not show how many times in the long run each of the possible outcomes should be guessed.

6

CHAPTER XXXIII.

OBSERVATION AND MEMORY.

INDUCTION tries to weave facts together into a coherent world. But our knowledge of every one of these facts depends sooner or later upon a perception through the senses; and if our senses deceive us and we perceive or think we perceive what is not really present, that false perception will tend to give us a wrong conception of the world. Hence it is necessary to know something about the difference between good and bad perceptions. Moreover many of the perceptions from which we draw inferences took place some time ago, and if we depend upon our memory but do not remember them correctly, we are as badly off as if the perceptions themselves were wrong. Hence we must consider memory also.

Observation

ence.

The first thing to learn about Observation is the vast difference between what one actually perceives and the inference by which he explains it. The word 'Observation' seems to refer to the perception only; but and inferas it is generally used it includes a vast amount of inference also. To explain this difference. A Frenchman makes a flying visit to the United States and then goes home to write a book in which he recounts his observations upon the character of the American people. But the 'observations' he recounts involve at least three successive sets of inferences. What he has really observed is a specific set of words and acts on the part of this, that, and the other specific indi