2
$\begingroup$

I have a probability of an event $=p$. For example ($p = 0.01$). I have $N$ (say $1000$) independent trials. I seek to calculate the probability of having exactly $k$ ($k= 1 \dots N$) events of in the set.

I have that the probability of having $0$ events is $(1-p)^N$, which is to say the probability of getting no event $(1-p)$ in each of the $N$ trials.

After that I am not sure where to go. Simulations give very reasonable answers - i.e. a more or less Gaussian centered at 10 (1% of 1000 using the examples given). But I am stumped for a closed form solution.

$\endgroup$
2
$\begingroup$

In your example, where the probability of an individual event is small and the expected number of events is fairly small, a Poisson distribution will be a better approximation. There is a "closed form" for the cumulative distribution function so you don't have to do the sum if you have the incomplete gamma function available to you. The probability of at most $k$ events is $\frac {\Gamma(k+1,\lambda)}{k!}$ where $\lambda=Np$ is the expected number of events. The probability of exactly $k$ events is $\frac {\lambda^k}{k!}e^{-\lambda}$. The Poisson distribution becomes Gaussian as the number of expected events becomes large.

$\endgroup$
  • $\begingroup$ Thanks, I am doing the analysis in Excel and there appears to be a way to get the incomplete gamma function. I will try. Thanks. $\endgroup$ – Gary Apr 19 '13 at 3:25
1
$\begingroup$

Probability of having exactly $k$ events (occurence probability $p$) out of $N$ is

$$ \binom{N}{k} p^{k} (1-p)^{N-k}$$

See Binomial Distribution (and that indeed can be approximated with the Gaussian).

$\endgroup$
  • $\begingroup$ Thanks, the form you provided gives the probability for exactly k, so if I sum from 1 to k I will get the answer I need. Probably the integral to the Gaussian would give a good approximation. $\endgroup$ – Gary Apr 19 '13 at 2:39
  • $\begingroup$ You can aproximate the binomial distribution to normal only when N is big (in this case, it is) and p is 'far' from 0 or 1. Take a look at: en.wikipedia.org/wiki/… $\endgroup$ – Rcoster Apr 19 '13 at 12:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.