The Laws of Probability

The science of probabilities furnishes three principles of which practical use is made in life insurance. They may be called respectively (1) the law of certainty, (2) the law of simple probability, and (3) the law of compound probability. Their use makes possible the description of risk in terms of mathematical values, and the statement of the three laws is as follows: (1) certainty may be expressed by unity, or one; (2) simple probability, or the probability or chance that an event will happen or that it will not happen may be expressed by a fraction; and (3) compound probability, or the chance that two mutually independent events will happen is the product of the separate probabilities that the events, taken separately, will happen.

An illustration will serve to make these statements clear. If a box contains twenty marbles and it is known that five of the marbles are black and the remainder white, let us suppose it is desired to know the probability that a marble drawn at random from the box will be black. If any marble has equal chances with any other of being drawn, then there are twenty different draws that might be made and if five of the marbles are black then it can be said that there are five chances out of twenty of drawing a black marble, or the chance is in the ratio of five to twenty, or is 5/20. This fraction is obtained in the following manner: The denominator equals the total number of marbles in the box; the numerator equals the number that satisfies the condition stated, namely, the quality of being black. In like manner it might be desired to know the chance that the marble will not be black, and by a like method of reasoning it is found that this probability equals 15/20. From these facts it is possible to formulate a general statement of the method of determining simple probabilities as follows: The denominator will equal the total number of possible trials or chances that a thing may happen or may not happen or the total number of instances dealt with - in the example above, total marbles. The numerator will be composed of those instances only which satisfy the conditions imposed - in the same example, black marbles.

In the illustration here used there are marbles of two kinds only, black and white, and any marble withdrawn from the box must be one or other color. The total existing probabilities are therefore two, the probability of drawing a black marble and the probability of drawing a white one. If certainty is represented by unity, then unity, or the value "one", will represent the fact of drawing any marble. But any marble drawn at random may be either black or white and since the probability of drawing the former is 5/20, and of the latter 15/20, and since certainty must equal the sum of all equals 1, therefore certainty must equal the sum of all separate probabilities, in this case 5/20 + 15/20 = 1.

This corollary that certainty equals the sum of all separate probabilities may be further illustrated by the familiar example of the coin. It is certainty that a coin tossed into the air will come to rest on one side and this fact is represented by the value "one". Now, since it has but two sides, the sum of the separate probabilities that it will alight heads up or tails up must equal one. The probability of falling heads up, determined by the above rule for valuing simple probabilities, is ½, since there are two possible sides and one is heads; likewise the probability of falling tails up is ½, and the sum of these two fractions equals one.

The probability that both of two mutually independent events will happen is equal to the product of the simple probabilities that the events taken separately will happen. Suppose that two coins are tossed up and it is desired to know the chance that they will both fall heads up. By the statement of the law above it will be ½ x ½ or ¼, since it is known that the chance is ½ that each separate coin will fall heads up. That this is the correct result may easily be demonstrated. Suppose the two coins are a nickel and a dime. Then the different ways in which they may fall are:

These four combinations comprise the only possible ones that can be made with the two coins and the first combination is the only one of the four that satisfies the stipulated conditions, namely, both coins heads up. Hence there is one chance in four for this combination to appear, or the probability of its occurrence is ¼.

According to the law of compound probabilities, as stated herewith, the product of simple probabilities equals the probability that two events will happen at the same time, only when the two events are mutually independent. The happening of the one must have no effect upon the occurrence or non-occurrence of the other, that is, must neither make it necessary for the second to occur nor make it impossible. If the law were valid irrespective of this qualification, such absurd results as the following might be obtained. The chance that the coin will fall heads up is ½ and the chance that it will fall tails up is likewise ½. Therefore the chance that it will fall both heads up and tails up is ½ X ½ or ¼. The absurdity results from the fact that the occurrence of the first named event makes it impossible for the second to occur simultaneously.