Suppose we flip a coin until it lands heads. What is the probability that we see a head after one flip? Two flips? $n$ flips? We want to extend our notion of probability distribution to a set with infinite elements, but only countably many.

Consider a set $A$.^{1}
If $A$ has $n$ elements, then a probability
distribution on $A$ is $p: A \to \R $ where
$p(a) = 1/n$.
There is a natural candidate.

What if $A$ is the set of natural numbers $\N $. The principle difficulty is that not all sequences of real numbers $a: \N \to \R $ are summable.

A (countable) probability distribution on $\N $ is $p: \N \to \R $ where $p \geq 0$ and

\[ \textstyle \sum_{n =1}^{\infty} p(n) = 1. \]

More generally, a probability distribution on some countable set $C$ which we have numbered $c: \N \to C$, is a function $p: C \to [0,1]$ such that

\[ \sum_{i = 1}^{n} p(c_i) = 1 \]

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