Page:Elementary algebra (1896).djvu/349

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