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 CHAPTER XXXVI.

Probability (Chance).

402. Definition. If an event can happen in $$a$$ ways and fail in $$b$$ ways, and each of these ways is equally likely, the probability, or the chance, of its happening is $a⁄a+b$, and of its failing is $b⁄a+b$. Hence to find the probability of an event happening, divide the number of favorable ways by the whole number of ways favorable and unfavorable.

For instance, if in a lottery there are 7 prizes and 25 blanks, the chance that a person holding 1 ticket will win a prize is $7⁄32$, and his chance of not winning is $25⁄32$.

Instead of saying that the chance of the happening of an a event is $a⁄a+b$, it is sometimes stated that the odds are $$a$$ to $$b$$ in favor of the event, or $$b$$ to $$a$$ against the event.

Thus in the above the odds are seven to twenty-five in favor of the drawing of a prize, and twenty-five to seven against success.

403. The reason for the mathematical definition of probability may be made clear by the following considerations:

If an event can happen in $$a$$ ways and fail to happen in $$b$$ ways, and all these ways are equally likely, we can assert that the chance of its happening is to the chance of its failing as $$a$$ to $$b$$. Thus if the chance of its happening is represented by $$ka$$, where $$k$$ is an undetermined constant, then the chance of its failing will be represented by $$kb$$. ∴ chance of happening + chance of failing = $$k(a + b)$$. 331