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 0 to some number. Hence it follows that in the theory of probability certainty is expressed by 1, and impossibility by 0; 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 1/2 × 1/2, or 1/4; the probability of throwing it three times running is 1/2 × 1/2 × 1/2, or 1/8; 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 1/2, and at the second trial also 1/2, the probability of throwing it in the first two throws is 1/2 + 1/2, or certainty. Not only is this result evidently absurd, but a repetition of the process would lead us to a probability of 1 1/2 or of any greater number, results which could have no meaning whatever. The probability we wish to calculate is that of one head in two throws, but in our addition we have included the case in which two heads appear. The true result is 1/2 + 1/2 × 1/2 or 3/4, or the probability of head at the first throw, added to the exclusive probability that if it does not come at the first, it will come at the second. The greatest difficulties of the theory arise from the confusion of exclusive and unexclusive alternatives. I may remind the reader that the possibility of unexclusive alternatives was a point previously discussed (p. [68]), and to the reasons then given for considering alternation as logically unexclusive, may be added the existence of these difficulties in the theory of probability. The erroneous result explained above really arose from overlooking the fact that the expression “head first throw or head second throw” might include the case of head at both throws.
The Logical Alphabet in questions of Probability.
When the probabilities of certain simple events are given, and it is required to deduce the probabilities of compound events, the Logical Alphabet may give assistance, provided that there are no special logical conditions so that all the combinations are possible. Thus, if there be three events, A, B, C, of which the probabilities are, α, β, γ, then the negatives of those events, expressing the absence of the events, will have the probabilities 1 - α, 1 - β, 1 - γ. We have only to insert these values for the letters of the combinations and multiply, and we obtain the probability of each combination. Thus the probability of ABC is αβγ; of Abc, α(1 - β)(1 - γ).
We can now clearly distinguish between the probabilities of exclusive and unexclusive events. Thus, if A and B are events which may happen together like rain and high tide, or an earthquake and a storm, the probability of A or B happening is not the sum of their separate probabilities. For by the Laws of Thought we develop A ꖌ B into AB ꖌ Ab ꖌ aB, and substituting α and β, the probabilities of A and B respectively, we obtain α . β + α . (1 - β) + (1 - α) . β or α + β - α . β. But if events are incompossible or incapable of happening together, like a clear sky and rain, or a new moon and a full moon, then the events are not really A or B, but A not-B, or B not-A, or in symbols Ab ꖌ aB. Now if we take μ = probability of Ab and ν = probability of aB, then we may add simply, and the probability of Ab ꖌ aB is μ + ν.
Let the reader carefully observe that if the combination AB cannot exist, the probability of Ab is not the product of the probabilities of A and b. When certain combinations are logically impossible, it is no longer allowable to substitute the probability of each term for the term, because the multiplication of probabilities presupposes the independence of the events. A large part of Boole’s Laws of Thought is devoted to an attempt to overcome this difficulty and to produce a General Method in Probabilities by which from certain logical conditions and certain given probabilities it would be possible to deduce the probability of any other combinations of events under those conditions. Boole pursued his task with wonderful ingenuity and power, but after spending much study on his work, I am compelled to adopt the conclusion that his method is fundamentally erroneous. As pointed out by Mr. Wilbraham,[113] Boole obtained his results by an arbitrary assumption, which is only the most probable, and not the only possible assumption. The answer obtained is therefore not the real probability, which is usually indeterminate, but only, as it were, the most probable probability. Certain problems solved by Boole are free from logical conditions and therefore may admit of valid answers. These, as I have shown,[114] may be solved by the combinations of the Logical Alphabet, but the rest of the problems do not admit of a determinate answer, at least by Boole’s method.