# n agents accessing a resource

I want to solve a simple stochastic problem. Imagine there are n agents who want to access a resource, with a probability p at a given time t. What ist the probability that the resource will be free and not accessed by an agent. How does the result change with 2n and 2 available resources? How do i solve this problem?

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Could you walk us through a simple example for the case of n=2, i.e. 2 agents with 1 resource, and 4 agents with 2 resources? – Marcus Fry Oct 29 '10 at 18:02
Huh ok, pfff let's imagin you have 2 roomates and a video game console, you know that the other two players want to play a video game with the probability of 0.1 at a time t (for the porpose of the problem, let's assume that only one person can play at a time). So what is the probability that you can play at time t. The 2n is obvious. Now you have 5 roommates and 2 consoles. What's the probability of accessing any one of them. – inf.ig.sh Oct 29 '10 at 18:55

The resource will be free with probability $(1-p)^n$.
When there are $2n$ agents and $2$ resources, one of the resources will be free with probability $(1-p)^{2n} + 2np(1-p)^{2n-1}$.
In general, you want to calculate the tail of a Binomial distribution. When $p = c/n$ and $n$ is large, the distribution is approximately Poisson.