y. But these two last expectation is y. But these two last expectations together are evidently the same with my original expectation, the value of which is a and therefor P+y=r. But y is to z as n-b is to n. wherefore a is to P as N is to b, and *(the probability of my obtaining N)is 5 Cor. Suppose after the expectation given me in the foregoing proposition and before it is at all known whether the lst event has happened or not, I should find that the 2d event is determined on which my expectation depended, and have no reason to esteem the value of my expectation either greater or less, it would be reasonable for me to give something to be reinstated in my former circumstances, and this over and over again as i should be informed that the 2d event had happened, which is evidently absurd. And the like absurdity plain follows if you say I ought to set a greater value on my expectation than before for then it would be reasonable for me to refuse something if offered me upon ondition I would relinquish it, and be reinstated in my former circumstances and this likewise over and over again as often as(nothing being known concern ing the Ist event)it should appear that the 2d had happened. Notwithstanding therefore that the 2d event has happened, my expectation ought to be esteemed the same as before i. e. I, and consequently the probability of my obtaining N is(by definition 5)still f or F. But after this discovery the probability of my obtaining N is the probability that the lst of two subsequent events has happen=ed upon the supposition that the 2d has, whose probabilities were as before specified. But the probability that an event has happened is the same as as the probability i have to guess right if i guess it has happened. Wherefore the following proposition is evident PROP 5 If there be two subsequent events, the probability of the 2d n and the probability of both together f, and it being lst discovered that the 2d ev has slso happened, the probability I am right is t What is here said may perhaps be a little illustrated by considering that all that can be lost by the happening of the 2d event is the chance I should have of being reinstated in my formed circumstances, if the event on which my expectation depended had been determined in the manner expressed in the propostion. But this chance is always as much agains is for me. If the Ist event happens, it is against me, and equal to the chance for the 2d vent's failing. If the lst event does not happen, it is for me, and equal also to the chance fo the 2d event's failing. The loss of it, therefore, can be no disadvantage t What is proved by Mr. Bayes in this and the preceding proposition is the same with the answer to the following question. What is the probability that a certain event, when it s one of the events is given, nothing can be due for the expectation of it; and, consequently, the value of an expectation depending on the happening of both events must be the sam with the value of an expectation depending on the happening of one of them. In other with the probability of this other. Call r then the probability of this other, and if f be the probability of the given event, and f the probability of both, because f=X
y. But these two last expectation is y. But these two last expectations together are evidently the same with my original expectastion, the value of which is x, and therefor P + y = x. But y is to x as N − b is to N. Wherefore x is to P as N is to b, and x N (the probability of my obtaining N) is P b . Cor. Suppose after the expectation given me in the foregoing proposition, and before it is at all known whether the 1st event has happened or not, I should find that the 2d event is determined on which my expectation depended, and have no reason to esteem the value of my expectation either greater or less, it would be reasonable for me to give something to be reinstated in my former circumstances, and this over and over again as I should be informed that the 2d event had happened, which is evidently absurd. And the like absurdity plainly follows if you say I ought to set a greater value on my expectation than before, for then it would be reasonable for me to refuse something if offered me upon condition I would relinquish it, and be reinstated in my former circumstances; and this likewise over and over again as often as (nothing being known concerning the 1st event) it should appear that the 2d had happened. Notwithstanding therefore that the 2d event has happened, my expectation ought to be esteemed the same as before i. e. x, and consequently the probability of my obtaining N is (by definition 5) still x N or P b ∗. But after this discovery the probability of my obtaining N is the probability that the 1st of two subsequent events has happen=ed upon the supposition that the 2d has, whose probabilities were as before specified. But the probability that an event has happened is the same as as the probability I have to guess right if I guess it has happened. Wherefore the following proposition is evident. P R O P. 5. If there be two subsequent events, the probability of the 2d b N and the probability of both together P N , and it being 1st discovered that the 2d event has slso happened, the probability I am right is P b †. ∗What is here said may perhaps be a little illustrated by considering that all that can be lost by the happening of the 2d event is the chance I should have of being reinstated in my formed circumstances, if the event on which my expectation depended had been determined in the manner expressed in the propostion. But this chance is always as much against me as it is for me. If the 1st event happens, it is against me, and equalto the chance for the 2d event’s failing. If the 1st event does not happen, it is for me, and equalalso to the chance for the 2d event’s failing. The loss of it, therefore, can be no disadvantage. †What is proved by Mr. Bayes in this and the preceding proposition is the same with the answer to the following question. What is the probability that a certain event, when it happens, will be accompanied with another to be determined at the same time? In this case, as one of the events is given, nothing can be due for the expectation of it; and, consequently, the value of an expectation depending on the happening of both events must be the same with the value of an expectation depending on the happening of one of them. In other words; the probability that, when one of two events happens, the other will, is the same with the probability of this other. Call x then the probability of this other, and if b N be the probability of the given event, and p N the probability of both, because p N = b N × x, x = p b = the probability mentioned in these propositions. 6
PROP. 6 The probability that several independent events shall happen is a ratio com- pounded of the probabilities of each For from the nature of independent events, the probability that any one happens is not altered by the happening or gailing of any one of the rest, and consequently the probability that the 2d event happens on supposition the lst does is the same with its original probability; but the probability that any two events happen is a ratio compounded of the lst event, and the probability of the ed on the supposition on the lst happens by prop. 3. Wherefore the probability that any two independent events both happen is a ratio compounded of the lst and the probability of the 2d. And in the like manner considering the lst and events together as one event: the probability that three independent events all happen is a ratio compounded of the probability that the two lst both happen ity of the 3d. And thus you may proceed if there be ever so many such events; from which the proposition is manifest Cor. 1. If there be several independent events, the probability that the lst happens the 2d fails, the 3d fails and the 4th happens, &c is a ratio compounded of the probability of the lst, and the probability of the failure of 2d, and the probability of the failure of the 3d, and the probability of the 4th, &c. For the failure of an event may always be considered as the happening of its contrary Cor. 2. If there be several independent events, and the probability of eac one be a, and that of its failing be b, the probability that the lst happens and the 2d fails, and the 3d fails and the 4th happens, &c. will be abba, &zc. For accor ding to the algebraic way of notation, if a denote any ratio and b another abba denotes the ratio compounded of the ratios a, b, b, a. This corollary is therefore only a particular case of the foregoing Definition. If in consequence of certain data there arises a probability that a certain event should happen, its happening or failing, in consequence of these data, I call it's happening or failing in the lst trial. And if the same data be again repeated, the happening or failing of the event in consequence of them call its happening or failing in the 2d trial; and so again as often as the same data are repeated. And hence it is manifest that the happening or failing of the same event in so many differe- trials, is in reality the happening or failing of so many distinct independent events exactly similar to each other PROP. 7 If the probability of an event be a, and that of its failure be b in each single trial, the probability of its happening p times, and failing g times in p+q trials is Eapbq ife be the coefficient of the term in which occurs aPbq when the binomial a+66+q is expanded For the happening or failing of an event if different trials ind pendent events. Wherefore(by cor. 2. prop. 6. the probability that the event happens the lst trial, fails the 2d and 3d, and happens the 4th, fails the 5th. &c. (thus happening and failing till the number of times it happens be p and the
P R O P. 6. The probability that several independent events shall happen is a ratio compounded of the probabilities of each. For from the nature of independent events, the probability that any one happens is not altered by the happening or gailing of any one of the rest, and consequently the probability that the 2d event happens on supposition the 1st does is the same with its original probability; but the probability that any two events happen is a ratio compounded of the 1st event, and the probability of the 2d on the supposition on the 1st happens by prop. 3. Wherefore the probability that any two independent events both happen is a ratio compounded of the 1st and the probability of the 2d. And in the like manner considering the 1st and 2d events together as one event; the probability that three independent events all happen is a ratio compounded of the probability that the two 1st both happen and the probability of the 3d. And thus you may proceed if there be ever so many such events; from which the proposition is manifest. Cor. 1. If there be several independent events, the probability that the 1st happens the 2d fails, the 3d fails and the 4th happens, &c. is a ratio compounded of the probability of the 1st, and the probability of the failure of 2d, and the probability of the failure of the 3d, and the probability of the 4th, &c. For the failure of an event may always be considered as the happening of its contrary. Cor. 2. If there be several independent events, and the probability of each one be a, and that of its failing be b, the probability that the 1st happens and the 2d fails, and the 3d fails and the 4th happens, &c. will be abba, &c. For, according to the algebraic way of notation, if a denote any ratio and b another abba denotes the ratio compounded of the ratios a, b, b, a. This corollary is therefore only a particular case of the foregoing. Definition. If in consequence of certain data there arises a probability that a certain event should happen, its happening or failing, in consequence of these data, I call it’s happening or failing in the 1st trial. And if the same data be again repeated, the happening or failing of the event in consequence of them I call its happening or failing in the 2d trial; and so again as often as the same data are repeated. And hence it is manifest that the happening or failing of the same event in so many differe- trials, is in reality the happening or failing of so many distinct independent events exactly similar to each other. P R O P. 7. If the probability of an event be a, and that of its failure be b in each single trial, the probability of its happening p times, and failing q times in p+q trials is E apbq if E be the coefficient of the term in which occurs apbq when the binomial a + b| b+q is expanded. For the happening or failing of an event if different trials are so many independent events. Wherefore (by cor. 2. prop. 6.) the probability that the event happens the 1st trial, fails the 2d and 3d, and happens the 4th, fails the 5th. &c. (thus happening and failing till the number of times it happens be p and the 7
number it fails be g) is abbab &c. till the number of a' s be p and the number of b's be g, that is:'tis aPb9. In like manner if you consider the event as happening p times and failing q times in any other particular order, the probability for it ab: but the number of different orders according to which an event may happen or fails so as in all to happen p times and fail g, in p+q trials is equal to the number of permutations that aaaa bbb admit of when the number of a's is p and the number of b s is g. And this number is equal to e, the coefficient of the term in which occurs aPba when a+bp+q is expanded. The event therefore may happen p times and fail g in p+q trials e different ways and no more, and its happening and failing these several different ways are so many inconsistent events, the probability for each of which is aPb, and therefore by prop. 1. the probability that some way or other it happens p times and fails q times in p+q trials is e aPbq SECTION I lere postulate. 1. Suppose the square table or plane abcd to be so made and yelled, that if either of the balls o or W be thrown upon it, there shall be the same probability that it rests upon any one equal part of the plane as another, and that it must necessarily rest somewhere upon it that the ball w shall be lst thrown, and through the where it rests a line os shall be drawn parallel to AD, and meeting CD and AB in s and o; and that afterwards the ball O shall be thrown p+g or n times, and that its resting between AD and os after a single throw be called the happening of the event M in a single trial. These things supposed. Lem. 1. The probability that the point o will fall between any two points the line AB is the ratio of the distance between the two points to the whole line aB Let any two points be named, as f and b in the line AB, and through them parallel to Ad draw fF, bl meeting CD in F and L. Then if the rectangles Cf, b, LA are commensurable to each other, they may each be divided into the same equal parts, which being done, and the ball w thrown, the probability it will rest somewhere upon any number of these equal parts will be the sum of the probabilities it has to rest upon each one of them, because its resting upon any different parts of the plance AC are so many inconsistent events; and this sum, the same, is the probability it should rest upon any one equal part multiple es because the probability it should rest upon any one equal part as another is oy the number of parts. Consequently, the probability there is that the ball W should rest somewhere upon Fb is the probability it has to rest upon one equal part multiplied by the number of equal parts in Fb; and the probability it rests somewhere upon Cf or LA, i.e. that it dont rest upon Fb(because it must rest somewhere upon AC) is the probability it rests upon one equal part multiplied by the number of equal parts in Cf, La taken together. Wherefore the probability it rests upon Fb is to the probability it dont as the number of equal parts in Fb is to the number of equal parts in Cf, La together, or as Fb to Cf, LA together, or as fb to Bf, Ab together. And(compend inverse) the
number it fails be q) is abbab &c. till the number of a’s be p and the number of b’s be q, that is; ’tis apbq. In like manner if you consider the event as happening p times and failing q times in any other particular order, the probability for it is apbq; but the number of different orders according to which an event may happen or fails so as in all to happen p times and fail q, in p + q trials is equal to the number of permutations that aaaa bbb admit of when the number of a’s is p and the number of b’s is q. And this number is equal to E, the coefficient of the term in which occurs apbq when a + b| p+q is expanded. The event therefore may happen p times and fail q in p + q trials E different ways and no more, and its happening and failing these several different ways are so many inconsistent events, the probability for each of which is apbq, and therefore by prop. 1. the probability that some way or other it happens p times and fails q times in p + q trials is E apbq. S E C T I O N II. Postulate. 1. Suppose the square table or plane ABCD to be so made and levelled, that if either of the balls o or W be thrown upon it, there shall be the same probability that it rests upon any one equal part of the plane as another, and that it must necessarily rest somewhere upon it. 2. I suppose that the ball W shall be 1st thrown, and through the point where it rests a line os shall be drawn parallel to AD, and meeting CD and AB in s and o; and that afterwards the ball O shall be thrown p + q or n times, and that its resting between AD and os after a single throw be called the happening of the event M in a single trial. These things supposed, Lem. 1. The probability that the point o will fall between any two points in the line AB is the ratio of the distance between the two points to the whole line AB. Let any two points be named, as f and b in the line AB, and through them parallel to AD draw fF, bL meeting CD in F and L. Then if the rectangles Cf, Fb, LA are commensurable to each other, they may each be divided into the same equal parts, which being done, and the ball W thrown, the probability it will rest somewhere upon any number of these equal parts will be the sum of the probabilities it has to rest upon each one of them, because its resting upon any different parts of the plance AC are so many inconsistent events; and this sum, because— the probability it should rest upon any one equal part as another is the same, is the probability it should rest upon any one equal part multiplied by the number of parts. Consequently, the probability there is that the ball W should rest somewhere upon Fb is the probability it has to rest upon one equal part multiplied by the number of equal parts in Fb; and the probability it rests somewhere upon Cf or LA, i.e. that it dont rest upon Fb (because it must rest somewhere upon AC) is the probability it rests upon one equal part multiplied by the number of equal parts in Cf, LA taken together. Wherefore, the probability it rests upon Fb is to the probability it dont as the number of equal parts in Fb is to the number of equal parts in Cf, LA together, or as Fb to Cf, LA together, or as f b to Bf, Ab together. And (compendo inverse) the 8