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) thenumber 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