A binomial experiment (Bin( $n$, $p$)) is made up of $n$ repeated independent Bernoulli experiments (also called binary trials) all with the same probability ( $p$) of success. The binomial events (random variables) are the total number of successes in $n$ trials (ranging from zero to $n$). It is precisely because the constituent Bernoulli trials are assumed to be independent that we can write down the binomial probability function as a simple product of the Bernoulli probabilities.

When $n$ becomes very large and $p$ very small, computing binomial probabilities becomes very tedious (because of the factorials and powers) so a useful approximation to the binomial distribution is the Poisson with parameter $\mathrm{np}$. The assumption of independence carries over because of the mathematical link with the binomial (proved in any statistics text). Independence is also at the root of the ''memoryless'' property of the Poisson process.