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 21/128. The probability of five feet three inches is 35/128; of five feet one inch 7/128; of five feet 1/128, and so on. Thus the eighth line of the Arithmetical Triangle gives all the probabilities arising out of the combinations of seven
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 7/24.
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.