MINISTRY OF CHANCE |
| DIRECTOR Rauno Koomis |
ABOUT
Let x denote the probability of happening of one event, y that of a second, z of a third1,
then then it will follow that 1—x, i—-y, 1— z, will represent the respective probabilities of their failing.
Let it be required to find the probability of the happening of them all, it is plain that by what has been already demonstrated, the answer will be x xy x z. If it be required to find the probability of their all failing, the answer will be 1—xx i—yx 1—z. Let it be proposed to assign the probability of some one of them, or more, happening. As this question is equivalent to this other, What is the probability of their not all failing, the an* swer will be 1—JZ^xT^x 1—*. Let it be demanded what is the probability of the happening of the first and second, and at the same time of the failing of the third, the answer will be xxyxTZ^Z. And for the same reason the probability of the happening of the second and third, and the failing of the first, will be y z x 1 —*. And again, for the probability of the happening of the first and third, and the failing of the second, the answer will be xzxTZTy, and the sum of these three probabilities, viz. xxyx y zx 1 —x-J-xzx 1 —y will express the probability of the happening of any two of them, but of no more than two.
C If
If it be demanded what is the probability of the happening of the first, to the exclusion of the other two, the answer will be x x i —y x i —z, and in the same manner the probability of the happening of the second, to the exclusion of the other two, will be y x i — *x i —x. Lastly, the probability of the happening of the third, to the exclusion of the other two, will be z x i —xx i—> The sum of these three probabilities, viz. x Xi x i — z-\-y x i —xx i — 2-j-zx i — xx i —>, will express the probability of the happening of any one of them, and the failing of the other two. In this manner may other casses of the same nature, belonging to any given number of events, be solved, several examples of which the reader will meet with in the ensuing pages.