may either happen or fail, that fraction will be a
proper designation of the probability of happening. Thus, if an event
has 3 chances to happen, and 2 to fail, then the fraction 3/5 will
fairly represent the probability of its happening, and may be taken to
be the measure of it.

The same may be said of the probability of failing, which will likewise
be measured by a fraction whose numerator is the number of chances
whereby it may fail, and the denominator the whole number of chances
both for its happening and failing; thus the probability of the failing
of that event which has 2 chances to fail and 3 to happen will be
measured by the fraction 2/5.

The fractions which represent the probabilities of happening and
failing, being added together, their sum will always be equal to
unity, since the sum of their numerators will be equal to their common
denominator. Now, it being a certainty that an event will either happen
or fail, it follows that certainty, which may be conceived under
the notion of an infinitely great degree of probability, is fitly
represented by unity.

These things will be easily apprehended if it be considered that the
word probability includes a double idea; first, of the number of chances
whereby an event may happen; secondly, of the number of chances whereby
it may either happen or fail. If I say that I have three chances to
win any sum of money, it is impossible from the bare assertion to
judge whether I am likely to obtain it; but if I add that the number of
chances either to obtain it or miss it, is five in all, from this will
ensue a comparison between the chances that are for and against me,
whereby a true judgment will be formed of my probability of success;
whence it necessarily follows that it is the comparative magnitude
of the number of chances to happen, in respect of the whole number
of chances either to happen or to fail, which is the true measure of
probability.

To find the probability of throwing an ace in two throws with a single
die. The probability of throwing an ace the first time is 1/6; whereof
1/ is the first part of the probability required. If the ace be
missed the first time, still it may be thrown on the second; but the
probability of missing it the first time is 5/6, and the probability of
throwing it the second time is 1/6; therefore the probability of missing
it the first time and throwing it the second, is 5/6 X 1/6 = 5/36 and
this is the second part of the proba