Probability of n events over a time period

[MrOd67]

TL;DR

How do you calculate the probability of n number of events over a time period given a constant probability p of the event occuring independently at any given time?

Let's say you have a leaking tab, and the probability of a droplet in any given second is 1%, regardless of whether there was a drop previously.

How would you calculate the probability of n drops in a minute?

No drops in a second is 0.99, so no drops over a minute is 0.99^60. Hence one or more drops is 1-0.99^60. But how about exactly n drops?

 

[PeroK]

The way you are modelling it you have a binomial distribution with effectively a trial every second and a independent probability of $p= 0.01$ of success in each and every trial.

https://en.wikipedia.org/wiki/Binomial_distribution

The probability of exactly $n$ drops involves binomial coefficients: $$P(n) = \binom{N}{n}p^n(1-p)^{N-n}$$ where $N = 60$ is the total number of trials.

You could model this sort of thing using a Poisson distribution:

https://en.wikipedia.org/wiki/Poisson_distribution

Although, for a dripping tap, maybe neither of these is quite right.

 

[MrOd67]

The way you are modelling it you have a binomial distribution with effectively a trial every second and a independent probability of $p= 0.01$ of success in each and every trial.

https://en.wikipedia.org/wiki/Binomial_distribution

The probability of exactly $n$ drops involves binomial coefficients: $$P(n) = \binom{N}{n}p^n(1-p)^{N-n}$$ where $N = 60$ is the total number of trials.

You could model this sort of thing using a Poisson distribution:

https://en.wikipedia.org/wiki/Poisson_distribution

Although, for a dripping tap, maybe neither of these is quite right.

Perfect, that's exactly what I was looking for. Thank you very much!