Basic probability question: 1 or more objects in N cells

[belliott4488]

Sorry that subject is vague. Here's the question:

If I have N objects randomly placed into M cells (M>N), what is the probability that I will find one or more objects in anyone cell?

Here's why I'm asking, or rather, here's why I'm confused:

I have one object that has been placed at random into one of 100 cells, it seems fair to say that if I pick anyone cell I will then have a 1/100 chance of finding the object in that cell. If I pick 5 random cells, I should then have a 5/100 chance of finding the object(?).

Now, what if I have ten objects placed at random into those 100 cells? (Right now I'll just say that a given cell may contain 1 or more objects - I'm not sure if that matters.) Is the probability of finding at least one object in a randomly selected cell then 1/10? I would think so, except that if I then pick more than one cell, I seem to get into trouble. What if I pick ten cells? Is the probability of finding at least one object then 1? What if I pick 12 cells? Probability is 1.2?

You see my problem. This is obviously a very basic question, but it's not the first time I've been tripped up by a basic question in prob/stats.

ooh, wait ... Now that I've typed this up, I think I might see my problem. What I'm asking is "What is the probability of finding an object in cell 1 OR in cell 2 OR in cell 3 etc." Since the probabilities of the objects being in different cells are independent, I shouldn't simply add the individual probabilities - I must also subtract off the probability of finding an object in cell 1 AND cell 2 etc (to compensate for double-counting the cases where I find an object in one cell). That might resolve my problem, but I'm going to post this anyway, in case it's instructive to anyone else (and also to get confirmation that I'm on the right track - or correction, if not).

Thanks for any help,
Bruce

 

[Valkarie]

Yes, you are correct. The probability of finding at least one object in a randomly selected cell is equal to 1 minus the probability of finding no objects in any of the cells. For example, if you have ten objects placed randomly into 100 cells, then the probability of finding at least one object in a randomly selected cell is equal to 1 - (99/100)10. This is because the probability of finding no objects in any of the cells is equal to the product of probabilities of finding no objects in each cell, which is (99/100) multiplied by itself 10 times.