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Two approaches with different answers where exactly 1 bucket is empty:

1: If the balls are labeled and the buckets are labeled, the sample space has size 256. Then there are 4 choices for the bucket with 2 balls, times 6 ways to choose those 2 balls, times 3 ways to choose the empty bucket, times 2 ways to choose where the 3rd ball goes, giving an event size of 144. Therefore, the probability is 144 divided by 256.

2: Now if the balls are unlabeled and the buckets are labeled, the sample space has size 7 choose 3, which is 35. Then there are 4 ways to pick the bucket with 2 balls times 3 ways to pick the empty bucket. Therefore, the probability is 12 divided by 35.

Why should labeling the balls change the probability that exactly one bucket is empty?

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  • $\begingroup$ Welcome to the math site. I’d encourage you to edit either your title or the body of your post because at present they disagree on exactly what it is that you’re asking. $\endgroup$ Nov 10, 2023 at 14:45
  • $\begingroup$ Of course labeling the balls does not change the probability. I encourage you for an experiment. $\endgroup$
    – user
    Nov 10, 2023 at 22:35
  • $\begingroup$ math.stackexchange.com/q/3411716/398708 $\endgroup$
    – Dan
    Nov 10, 2023 at 23:38

1 Answer 1

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It really has to do with how the balls are put into the bucket:

The first model assumes that each ball is put in independently and uniformly at random, so each ball is equally likely to occupy any bucket, while the second as assumes that each pattern cluster is equally likely.

Obviously the first is a more appropriate reflection of reality as we perceive it, "Stars and bars" is sometimes called Bose-Einstein statistics because bosons exhibit clustering, and is closer to the stars and bars model.

As a simple example, consider $10$ tosses of a ball uniformly at random between two "buckets" labelled $A$ and $B$. Stars and bars will give $11$ results $(10-0,9-1,8-2,...0-10$ so each would supposedly have equal probabilities of $\frac1{11}$, whereas if you actually perform an experiment, you will find that the $10-0$ pattern will have a probability of $\approx \frac1{1000}$ while a $5-5$ result will have a probability $\approx 250$ times higher !

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