# Basic probability question: 1 or more objects in N cells

• belliott4488
In summary: Therefore, the probability of finding at least one object is equal to 1 - (99/100)10. In summary, the probability of finding at least one object in a randomly selected cell when N objects are placed randomly into M cells (M>N) is equal to 1 - ((M-1)/M)N.
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

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.

## 1. What is the probability of getting at least one object in N cells?

The probability of getting at least one object in N cells is 1 - (1/N)^N. This formula is known as the "complement rule" in probability and is used when calculating the probability of an event not occurring.

## 2. How do you calculate the probability of getting exactly one object in N cells?

The probability of getting exactly one object in N cells is N*(1/N)^N. This formula takes into account the number of ways the event can occur and the probability of the event occurring in each way.

## 3. Is there a difference between getting at least one object and exactly one object in N cells?

Yes, there is a difference. Getting at least one object means that the event can occur in multiple ways, such as getting one object in one cell, two objects in two cells, or three objects in three cells. On the other hand, getting exactly one object means that the event can only occur in one specific way, such as getting one object in one cell and no objects in any other cells.

## 4. How does the probability change if the number of cells increases or decreases?

The probability of getting at least one object in N cells decreases as the number of cells increases. This is because there are more possible combinations of objects that can occur in a larger number of cells, making it less likely for at least one object to be present in all of them. Conversely, the probability increases as the number of cells decreases, as there are fewer possible combinations and therefore a higher chance of at least one object being present.

## 5. Can this concept be applied to real-life situations?

Yes, this concept of basic probability can be applied to real-life situations, such as the likelihood of winning a raffle or the chances of getting a certain number of correct answers on a multiple-choice test. It is also used in various fields of science, including genetics, where it is used to calculate the probability of certain traits being passed on from parents to offspring.

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