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6 THE DOCTRINE OF CHANCES.
the sum expected. Again, when the expectation depends on the happening of one event and the failure of another, then its value will be the product of the probability of the first happening, by the probability of the second failing, and that again by the value of the sum expected. This rule is applicable to the advent or not of as many events as may be assigned.
The above considerations apply to events which are independent; and in order to avoid any obscurity in the use of the terms, dependent and independent, we beg leave to define them.
Two events are independent when they have no connexion one with another, and that the happening of one has no influence upon the advent of the other. Two events are dependent when they are so connected that the probability of either happening is altered by the advent of the other.
From whence it may be inferred, that the probability of the happening of two events dependent, is the product of the probability of the advent 01 one of them by the probability which the other will have of arriving. This rule will extend to the happening of as many events as may be assigned.
But in the case of events dependent, to determine the probability of the advent of some of them, and at the same time the probability of the failing of some others, is a disquisition of greater difficulty, which will be more conveniently transferred to another place. 




