Naive Definition of Probability

The probability $P$ of an event $A$ is:

$P(A)=\frac{\text{\# favorable outcomes}}{\text{\# possible outcomes}}$

Warning

This assumes that all outcomes are equally likely and that you’ve got a finite sample space.

This works if, for example, you think about a fair coin or dice roll.

However, the definition breaks down outside of the most trivial examples. Intuitively, the probability of aliens existing on the moon isn’t $\frac{1}{2}$ (either they do, or they don’t); nor should the probability of blue aliens shaped specifically like a giraffe be equal to the probability of any aliens in general existing on the moon…it would be much less.