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Poisson distributions
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In the case where the probability is very small, the binomial distribution is reduced to a Poisson distribution.

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Distribución binomial
Equation

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Con la probabilidad de que se de un numero definido de pasos a la derecha e izquierda esta dada por

W_N(n_1,n_2)=\displaystyle\frac{N!}{n_1!n_2!}p^{n_1}q^{n_2}



con el número total de pasos es

N=n_1+n_2



y solo existe la probabilidad de ir a la derecha o a la izquierda, con se tiene para las probabilidades que

p+q=1



por lo que con se tiene la distribución binomial

W_N(n) =\displaystyle\frac{ N !}{ n !( N - n )!} p ^ n (1- p )^{ N - n }

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Application of the Sterling Approach
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Therefore expressions such as N!/(Nn)! for N large (N\gg 1) and n small (N\gg n) can be approximated with

u!\sim\sqrt{2\pi u}\left(\displaystyle\frac{u}{e}\right)^u



with what you get with N\gg n

\displaystyle\frac{N!}{(N-n)!}\sim\displaystyle\frac{\sqrt{2\pi N}}{\sqrt{2\pi (N-n)}}\displaystyle\frac{N^N}{(N-n)^{N-n}}\displaystyle\frac{e^{N-n}}{e^N}\sim N^n

that is

N^n\sim\displaystyle\frac{N!}{(N-n)!}

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Desviación Estandard Poison
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Estimate N! p^n/(N-n)! if p\sim 0 and N\gg n
Equation

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With the approximation

N^n\sim\displaystyle\frac{N!}{(N-n)!}



and employing

\lambda=Np



it can be shown that

\displaystyle\frac{N!}{(N-n)!}p^n\sim \lambda^n

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Estimate of (1-p)^{N-n} if p\sim 0 and N\gg n
Equation

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How the exponential is defined as

e^z\sim\left(1+\displaystyle\frac{z}{u}\right)^u



and by entering

\lambda=Np



you can replace z=-\lambda=-Np and u=N-n with N\gg n what results

e^{-\lambda}\sim (1-p)^{N-n}

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Probability for large N and small p
Equation

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Since the probability of taking n steps in one direction is

W_N(n) =\displaystyle\frac{ N !}{ n !( N - n )!} p ^ n (1- p )^{ N - n }



for a large number N and the probability is very small p \ll 1 can be approximated

\displaystyle\frac{N!}{(N-n)!}p^n\sim \lambda^n



and

e^{-\lambda}\sim (1-p)^{N-n}



the binomial distribution is reduced to a Poisson distribution:

P_{\lambda}(n) =\displaystyle\frac{ \lambda ^ n }{ n! }e^{- \lambda }

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Example comparison with Poisson distribution
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If we study the binomial distribution for large numbers N and very small probability p \ ll 1 it can be approximated using a Poisson distribution. The comparison can be done with the following simulator:

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Video: Poisson distributions