PROBABILITIES OF CAUSES I2. Direct and Inverse Probability.—Questions as to the respective probabilities of a specified event having been due to various possible causes have in the past produced a good deal of difficulty, and have been placed in a separate category as questions of inverse probability; questions as to probabilities of a specified cause leading to different events are then described as questions of direct probability. That difficulties should have arisen is not to be wondered at, as no clear definition of the prob ability of a cause seems to have been given: until we know what we mean by "probability," we cannot give a numerical value to it. A more serious cause of difficulty, in many cases, was the absence of sufficient information as to the antecedent proba bilities of the causes themselves. The final death-blow to the classification of questions of probability under the heads of "di rect" and " inverse" was given by the statistical treatment of correlation, which led to the study of probabilities of concurrent phenomena. What is the probability that a tall man has a tall wife? What is the probability that a boy who is good at arithmetic is good at composition? What is the probability that a girl with fair hair has blue eyes? Questions such as these found no place in the earlier classification. These difficulties have been met by the adoption of a statistical basis for the measurement of probability.
13. Typical Example.—The following may be taken as a typical question in inverse probability. A box contains 5 balls, and it is known that 1 of the balls is white and 3 are red, and that the remaining one is either white or red. A ball is drawn and is found to be white. What is the probability that the box con tained 2 white and 3 red balls? It is impossible to answer this question without further information. The box has been filled in some way: the method of filling having led to the box contain ing 1 white and 3 red balls, we require to know the probability that it has led to the other ball being white or red, respec tively. When we know this, the problem is a straightforward one.
Let us call the box a " W-box" or an " R-box" according as the other ball is white or red; and let us suppose that the prob abilities, or relative frequencies (sec. 4), of W-boxes and of R
boxes are in the ratio of 3:2. Then the problem can be stated as follows. There are W-boxes containing 2 white and 3 red balls, and R-boxes containing i white and 4 red balls; their frequencies being in the ratio of 3:2. A box is taken at random; and a ball is drawn from it and is found to be white. What is the proba bility that the box is a W-box? The ordinary method of answering the question is an appli cation of the rule that, if an event E has happened which may be due to any one of the causes Ci, ., and if the a priori probability of occurrence of the cause Cf is Pf, and if whenever Cf occurs the probability that it will lead to E is then the probability that E is due to Cf is proportional to PfPf, i.e., is Pfpf/(Pipi+P2p2+ ...). In the present case we have Pi =3/5, P2= 2/5, and pi= 2/5, P2= 1/5; whence PiPi:P2p2= 6:2 = 3:i, i.e., the probability that the box is a W-box is I.
For statistical treatment, a tabular arrangement is useful, as in Table II. This table shows a representative frequency distribution (sec. 15) for i,000 drawings. The figures in brackets indicate the order of entry in the table. Of the total i,000 boxes, 600 will (on the average) be W-boxes and 400 R-boxes. Of the 600 W-boxes, 240 (on the average) will give a white ball at a single drawing; of the 40o R-boxes, 8o will give a white ball. In the cases, therefore, in which a white ball is drawn, the prob ability of the box being a W-box is 240/(240+80) = I.
If the question had been " There are W-boxes containing 2 white and 3 red balls, and R-boxes containing 1 white and 4 red balls, their frequencies being in the ratio of ' 3 : 2 ; a box is taken at random and a ball is drawn from it; what is the prob ability that the ball will be white?" the question would have been a direct" one; but the one table would have been ap plicable to the two questions. We have already seen this in reference to Table I. (sec. 6); the same table enables us to answer the two questions, " What is the probability that a tall father has a tall son?" and " What is the probability that a tall son has a tall father?" though one of these might be regarded as a question of direct and the other of inverse probability.