A Universe of Sorts

Think about flipping a biased coin with some bias $p$ to associate a coin flip to each real number. Call this $b: \mathbb R \to \{0, 1\}$ .
Define the count of an interval $I$ as $\#I \equiv \{ r \in I | b(r) = 1 \}$ .
Suppose that this value $\#I$ is finite for any bounded interval.
Then the process we have is a poisson process.
Since the coin flips are independent, all 'hits' of the event must be independent.
Since there is either a coin flip or there is not, at most one 'hit' of the event can happen at any moment in time.
Since the bias of the coin is fixed, the rate at which we see $1$ s is overall constant.

§ Poisson Distribution