- #1

belliott4488

- 662

- 1

If I have N objects uniformly placed at random in a 1-d box of length b, how do I calculate the probability of finding one or more objects in a given length?

Here's what I mean:

I assume a uniform probability density of 1/b, that is, P = 1/b for 0<x<b and P = 0 everywhere else. I now place N objects into this region, according to that distribution. Thus I have a distribution of objects given by N/b for 0<x<b and zero everywhere else.

This seems to make sense, i.e. my total probability is 1, then total expected number of object in the box is N, and I can calculate the expected number in any segment of length a within the box as a*N/b.

What I am having trouble with is expressing the probability that I will find at least one object in a segment of length a. If a=b, then I'd better get 1 as my answer, but I haven't come up with the correct expression.

I guess I could also ask for the probability of finding n>=2, n>=3, ... n=N, as well, each of which should approach 1 as the length approaches b.

This feels like a pretty elementary question - is it?

Thanks for any suggestions.

Here's what I mean:

I assume a uniform probability density of 1/b, that is, P = 1/b for 0<x<b and P = 0 everywhere else. I now place N objects into this region, according to that distribution. Thus I have a distribution of objects given by N/b for 0<x<b and zero everywhere else.

This seems to make sense, i.e. my total probability is 1, then total expected number of object in the box is N, and I can calculate the expected number in any segment of length a within the box as a*N/b.

What I am having trouble with is expressing the probability that I will find at least one object in a segment of length a. If a=b, then I'd better get 1 as my answer, but I haven't come up with the correct expression.

I guess I could also ask for the probability of finding n>=2, n>=3, ... n=N, as well, each of which should approach 1 as the length approaches b.

This feels like a pretty elementary question - is it?

Thanks for any suggestions.

Last edited: