PROBLEM Given the number of times ion which an unknown event has happende and failed: Required the chance that the probability of its happening in a single trial lie somewhere between any two degrees of probability that can be named SECTION I DEFINITION 1. Several events are inconsistent, when if one of them hap- pens, none of the rest can 2. Two events are contrary when one, or other of them must; and both together cannot happer 3. An event is said to fail, when it cannot happen; or, which comes to the same thing, when its contrary has happened 4. An event is said to be determined when it has either happened or failed 5. The probability of any event is the ratio between the value at which expectation depending on the happening of the event ought to be computed, and the chance of the thing expected upon it's happening 6. By chance I mean the same as probability. 7. Events are independent when the happening of any one of them do neither increase nor abate the probability of the rest PROP 1 When several events are inconsistent the probability of the happening of one or other of them is the sum of the probabilities of each of them. Suppose there be three such events, and which ever of them happens I am receive N, and that the probability of the lst, 2d, and 3d are respectively N,N,N. Then(by definition of probability) the value of my expectation from the lst will be a. from the 2d 6. and from the 3d c. herefore the value of my expectations from all three is in this case an expectations from all three will be a+b+c. But the sum of my expectations from all three is in this case an expectation of receiving N upon the happening of one or other of them Wherefore(by definition 5)the probability of one or other of them is 4*+c or N+N+N. The sum of the probabilities of each of them Corollary. If it be certain that one or other of the events must happen, then a+b+c=n. For in this case all the expectations together amounting to a certain expectation of receiving n, their values together must be equal to N. And from hence it is plain that the probability of an event added to the probability of an event is f that of it's failure will be N-ps. Wherefore if the PROP 2 If a person has an expectation depending on the happening of an event, the probability of the event is to the probability of its failure as his loss if it fails to his gain if it happensP R O B L E M. Given the number of times ion which an unknown event has happende and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named. S E C T I O N I. DEFINITION 1. Several events are inconsistent, when if one of them hap￾pens, none of the rest can. 2. Two events are contrary when one, or other of them must; and both together cannot happen. 3. An event is said to fail, when it cannot happen; or, which comes to the same thing, when its contrary has happened. 4. An event is said to be determined when it has either happened or failed. 5. The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the chance of the thing expected upon it’s happening. 6. By chance I mean the same as probability. 7. Events are independent when the happening of any one of them does neither increase nor abate the probability of the rest. P R O P. 1. When several events are inconsistent the probability of the happening of one or other of them is the sum of the probabilities of each of them. Suppose there be three such events, and which ever of them happens I am to receive N, and that the probability of the 1st, 2d, and 3d are respectively a N , b N , c N . Then (by definition of probability) the value of my expectation from the 1st will be a, from the 2d b, and from the 3d c. Wherefore the value of my expectations from all three is in this case an expectations from all three will be a + b + c. But the sum of my expectations from all three is in this case an expectation of receiving N upon the happening of one or other of them. Wherefore (by definition 5) the probability of one or other of them is a+b+c N or a N + b N + c N . The sum of the probabilities of each of them. Corollary. If it be certain that one or other of the events must happen, then a + b + c = N. For in this case all the expectations together amounting to a certain expectation of receiving N, their values together must be equal to N. And from hence it is plain that the probability of an event added to the probability of its failure (or its contrary) is the ratio of equality. For these are two inconsistent events, one of which necessarily happens. Wherefore if the probability of an event is P N that of it’s failure will be N−P N . P R O P. 2. If a person has an expectation depending on the happening of an event, the probability of the event is to the probability of its failure as his loss if it fails to his gain if it happens. 4