Equally Likely Outcomes Distribution

Let there be a set $S$ with $N$ elements, each of them having the same probability. Then

$$P(S) = P\left({\,\bigcup_{i=1}^N E_i}\right)= \sum_{i=1}^N P(E_i)$$

$$= P(E_i) \sum_{i=1}^N 1 = NP(E_i).$$

Using $P(S) \equiv 1$ gives

$$P(E_i) = {1\over N}.$$

See also Uniform Distribution