Probability with urns and balls

• mynameisfunk
In summary, the conversation discusses how to calculate the probability of having a certain number of balls in the last urn when n balls are distributed into k urns. The approach involves finding the total number of possible outcomes by choosing a fixed number of balls in one urn and then distributing the remaining balls in different ways. The formula for combinations is also mentioned as a way to calculate the number of possible outcomes.
mynameisfunk

Homework Statement

If n balls are distributed into k urns, what is the probability that the last urn contains j balls?

The Attempt at a Solution

not really sure how to do it. This is only a review question, not HW, but with that being known, I would appreciate an explanation more than an answer.

Last edited:

There are a few ways to approach this problem. That being said..

To find to number of possible outcomes (i.e. ways to fill the urns), I would start by choosing a fixed number of balls that must be in the same urn, then finding out how many ways the others can be distributed.
So initially, let's say all n balls must be in one urn (clearly only one way to choose them). There are k urns to put them in, so this can be done in k ways.

Now, (n-1) balls must be in one urn. $${n\choose n-1}$$
so n ways to choose these n-1 balls. These can go in any urn, so k(n) ways to fill one urn with n-1 balls. The extra ball may go in any urn except the one already occupied, so it has k-1 urns to choose from. There are k(n)*(k-1) ways to fill the urns in this way.

Requiring that (n-2) balls be in one urn is similar, but the extra 2 balls may both go in any of the k-1 remaining urns, so they can be distributed in $$(k-1)^{2}$$ ways.

BTW: $$n\choose k$$ $$=\frac{n!}{k!(n-k)!}$$, if you aren't familiar with the notation. It's same thing as imposing an equivalence class so that order doesn't matter. I.E. (a, b) is the same thing as (b, a), which just means that you want the set {a, b}.

Does this make sense?

how would i calculate the total number of possible ways to put n balls in k urns?

1. What is the definition of probability in the context of urns and balls?

The probability in the context of urns and balls refers to the likelihood of a specific ball being selected from an urn containing a certain number of balls. It is expressed as a fraction or decimal between 0 and 1, where 0 represents impossibility and 1 represents certainty.

2. How do you calculate the probability of selecting a certain color ball from an urn?

The probability of selecting a certain color ball from an urn can be calculated by dividing the number of balls of that color by the total number of balls in the urn. For example, if an urn contains 10 red balls and 20 blue balls, the probability of selecting a red ball would be 10/30 or 1/3.

3. Can the probability of selecting a ball from an urn change?

Yes, the probability of selecting a ball from an urn can change if the number of balls in the urn changes. For example, if a ball is removed from the urn, the probability of selecting a different ball will increase.

4. How does the concept of replacement affect probability in urns and balls?

If balls are replaced after being selected from an urn, the probability of selecting a specific ball remains the same. However, if balls are not replaced, the probability of selecting a certain ball will change with each selection.

5. What is the difference between sampling with and without replacement?

Sampling with replacement means that balls are returned to the urn after being selected, while sampling without replacement means that balls are not returned to the urn. This affects the probability of selecting certain balls and can lead to different outcomes.

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