In many mathematical problems given as (natural language) text we are making assumptions. One example:
We roll two dices. What is the probability that a sum of 3 is thrown ?
By answering "$\frac{1}{18}$" we have assumed that two "fair" dices are used - although the text does not say if the dices are "fair" or "weighted". Using two "weighted" dices the result could be any number in the range $[0,1]$.
However we would definitely not assume that the probability of rolling the combination 1+2 (which can be rolled as 1+2 and as 2+1) is the same probability as the one of rolling the combination 1+1.
We would also not assume that all 11 possible sums from 2 to 12 have the same probability of $\frac{1}{11}$.
Against this background I have a problem understanding the assumptions made in questions with "indistinct objects" like this one:
There are 8 numbered cells and 12 indistinct balls. All 12 balls are randomly divided between all of the 8 cells. What is the probability ...
The answer is ... and it was confirmed by the official homework solution of the university.
If I understood correctly, the formula used in the official homework solution assumes that all "possible (distinguishable) results" in the process of dividing the balls between the cells have the same probability:
When two "indistinct" balls are divided between two (distinct) boxes the formula seems to result in the probability of $\frac{1}{3}$ for each of the three results (2+0, 1+1 and 0+2).
Comparing to the problem with the two dices however, I would expect that probabilities of $\frac{1}{4}$ are assumed for 2+0 and 0+2 and a probability of $\frac{1}{2}$ is assumed for 1+1.
Searching the internet for documents about the "indistinct balls in distinct boxes" problem I found out that the formula assumed by the homework solution is used by many universities.
In some of the answers and comments to the question linked there are comments like "I wonder if it is possible to simulate an experiment with this probability distribution at all".
My questions:
- Is there a reason why this probability distribution is assumed by different universities?
- Are there "real-life" examples (e.g. in science or in gambling) with "indistinct" objects that (approximately) have this probability distribution?
This is very strange. Could you give the precise problem text? Probabilities in this case should be 1/4, 1/2 and 1/4.
Here is the trick I use when reasoning about problems of this kind.
We have indistinguishable balls. One person makes experiments with these balls and cells. He puts balls randomly into cells, repeats the experiment many times and measures the frequency of different outcomes. He can't distinguish the balls.
Now suppose that there is another person sitting next to him. He has secretly attached very small lables on the balls, so he CAN distinguish the balls. And he also measures frequencies of different outcomes.
The second experimentator can see 4 different outcomes. First ball and second ball in cell 1, first ball in cell 1, second ball in cell 2, etc. Probabilities of these 4 outcomes are equal and it is 1/4.
This is not because of some convention!
Suppose 1000 experiments are taken simultaneously. First they put ball 1 into random cell. In about 500 experiments ball 1 is now in cell 1. Now we pay attention to these 500 experiments only. All them put ball 2 in a random cell. Again, in about half of experiments ball 2 will go into cell 1, and in the other half - into cell 2. "Half" - because placing ball 2 doesn't depend on what happened earlier. So we have that in about 250 of all the experiments ball 1 is in cell 1, and ball 2 is in cell 2. Situation with other possible outcomes is the same.
This may be not a rigorous prove that probabilities of all outcomes are equal. I wanted not to prove it, but to make it obvious.
Now back to the first experimentator - the one who can't distinguish the balls. It can't happen that he would see 1/3 probability of the 3 outcomes he can see. Because his results wouldn't agree with results of the second experimentator. Both of them have records of the same set of experiments. Second one would see that in about 1/4 of all experiments both balls are in cell 1, let's say that happened in experiments number 1, 8, 9, 19, 20,.... And the first experimentator would agree that in exactly these experiments both balls are in cell 1, so the probability of this outcome is 1/4.
Consider the following as offtopic:
Elementary particles ARE indistinguishable. It's not possible to put marks on them. Even theoretically. And if one puts two such indistinguishable particles into two boxes the probabilities of outcomes wouldn't be 1/4, 1/2, 1/4. This note was not about math, but about strange properties of the world we are living in.