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=Xy. 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 concern￾ing 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