I am struggling to understand a poisson distribution using my intuition of a binmomial distribution. Here is my logical flow:
Some quick nomenclature that I will be using:
An event is $e_0$, a successful outcome of an event is $e_+$, and a unit time is $t_0$. For example, an event could be a person seeing a result from a Google search, a successful outcome of the event is that person clicking on that link, and a unit time could be a minute.
The binomial distribution tells us the probability of having $x$ successes ($e_+$) in $n$ repeated events ($e_0$).
The Poisson distribution tells us the probability of having $x$ successes ($e_+$) occurring in a fixed interval of time ($t_0$).
Many texts refer to the $\lambda$ in the Poisson distribution as a rate. Thus, it would have units $\frac{e_+}{t_0}$.
Also, the expected value of the binomial distribution is $n*p = \lambda$.
I am struggling to integrate the concept of a "fixed interval of time" into my understanding of the binomial distribution since the binomial distribution doesn't really take into account the time it takes for an event to occur. I thought I would try and resolve this by adding units to the parameters. Thus, $$n*p = \lambda = \frac{n*e_0}{t_0}*\frac{p*e_+}{e_0} = \frac{\lambda*e_+}{t_0}$$.
Assuming this is OK to do, how do I then justify that $n$ needs to go to infinity and $p$ needs to go to 0 to convert the binomial to a poisson.