A binomial distribution is given by the probability mass function
\(\rm {p_X}\left( k \right) = P\left( {X = k} \right) ={}^nC_k\cdot{p^k}.{\left( {1 - p} \right)^{n - k}}\)
For large \(\rm n \gg k\) and small \(\rm p\left( {p \ll 1} \right)\), binomial distribution can be approximated as:
1
\(\rm P\left( {X = k} \right) \approx {e^{ - n}}.\frac{{{n^k}}}{{k!}}\)
2
\(\rm P\left( {X = k} \right) \approx {e^{ - p}}.\frac{{{n^k}}}{{k!}}\)
3
\(\rm P\left( {X = k} \right) \approx {e^{ - \left( {np} \right)}}.\frac{{{{\left( {np} \right)}^k}}}{{K!}}\)
4
\(\rm P\left( {X = k} \right) \approx {e^{ - p}}.\frac{{{p^k}}}{{K!}}\)