PROBABILITIES DERIVED FROM EXPERIENCE.
Probabilities of Causes.— Hitherto the de termination of the probability of an event has been based directly or indirectly upon certain governing events. A problem of another sort comes up for consideration. In the discussion of this problem and similar ones the word cause will be given an unusual meaning. For the present purposes the term will be used to mean a set of circumstances which might have given rise to an event. The problem may be stated as follows : It is known that an event E has happened, and it is known that one of reveal causes has given rise to the occurrence of it is required to determine the probability that E arose from a specified one of the causes.
A concrete example is furnished by the fol lowing problem.
Ex. 7. Two bags contain respectively one white and one black ball, and one white ball and nine black ones. A ball has been drawn from one of the bags, has been observed to be white and has been replaced. What is the probability that it was drawn from the first bag!' —Let p be the required probability. There are two ways of computing the probability of drawing a white ball from the first bag. In the first place it is necessary that the first bag should be selected and that, it having been selected, a white ball should be drawn. This gives it X = as a re sult. In the second place it is necessary that a white ball should be drawn and that, having been drawn, it should have come from the first bag. This yields (4 -I- • Hence, equating these values one gets p = 4.
Similar reasoning applied to the general case yields the following result: Theorem 4. If an event E has occurred as a result of one or another of certain causes C,, ..., Ca, whose respective probabilities pre vious to the occurrence of E were . • and if, when any one of the causes, as Ci, is known to be operating, the probability that it will produce E is pi, then the probability, Pi, that E resulted from Ci is Pi = ptirt -I- Pori + • • • + Apply the result to the following example: Ex_ 8. From a bag containing five balls, each of which is either white or black, four drawings have been made, the ball being re placed after each drawing. The result is three white balls and one black one. What are the
probabilities that the bag contains (1) four white, one black ball, (2) three white, two black, (3) two white three black, (4) one white, four blackr— there are four causes: (1) Four white, one black, (2) three white, two black, etc., and before the drawing they were equally probable. Hence ri=irir 4. Also Pi 1. Pi =(g)' Pa -= (Pt, P4= Then P, M. P2 _11, P. =4 P rig.
When the number of causes is very great the formula of theorem 4 becomes unwieldy. It is found that a simpler formula gives an approximate result. The method used in de riving this formula will be illustrated in the following example: Ex. 9. From among an exceedingly great number, N, of balls, each of which is either black or white, n balls are drawn, each ball being replaced before the next drawing. Of these exactly s are white. What are the proba bilities for the various possible numbers of white and black balls in the whole sett — Since before the drawings nothing was known as to the desired result, the a priori probabilities for all numbers of white balls from 0 to N are equal. Now the total number of white balls will be known if the ratio, r, of that number to N be known. Hence equally probable values of this ratio are all proper fractions having N as denominator. Since N is very great, r varies from 0 to 1 by very small increments. It will now be assumed that for purposes of approxi mation r may be treated as varying continuously from 0 to 1. Also, if r is the correct ratio, the probability that in n drawings s white and (n — s) black toillc would appear is in r*(1— r)a-8. If r varies continuously is —s from 0 to 1, the probability of any specified value of r is zero, but the a posteriori proba bility that the ratio shall lie between r and r + dr is — re (1— r)is—ocir — in 1 Pr 1 — r)o—•lr Is — s since the number of values of re (1 — r)n—• as r varies from r to r + dr is measured (see Geometri cal Probability) by dr; and the probability that r shall lie between r, and ri is In-I-1 Fs Ft A general theorem might be formulated em bodying this result, and such a theorem jointly with theorem 4 would constitute what is called the Theorem of Bayes.