Consider the following two problems:
- Two different balls are randomly distributed between two different urns. What is the probability of both balls being in the same urn?
- Two indistinguishable balls are randomly distributed between two different urns. What is the probability of both balls being in the same urn?
Both problems seem to have simple solutions:
- There are 4 elementary events: { ab | }, { a | b }, { b | a }, { | ab }. Two of them satisfy the composite event. All the elementary events have equal probabilities, thus we get an answer: 2/4 or 1/2.
- There are 3 elementary events: { ** | }, { * | * }, { | ** }. With the same reasoning here we get the probability of both balls being in the same urn equal 2/3.
To me it seems unnatural that in these two problems we get different answers. I do not see any practical reason why the type of balls would change probability we assign to both of them getting into the same urn.
After all, if, for example, their difference was in color, we could check their position blindfolded and still get results. So it should not matter to us if they are colored or not in the first place.
There is a clear correspondence between the two solutions:
- { ** | } ~ { ab | }
- { * | * } ~ { a | b }, { b | a }
- { | ** } ~ { | ab }
Then perhaps the events we considered elementary for the second solution are not such? The second event being composite and the answer still being 1/2?
It seems to me that this question boils down to the following: how to choose a sample space for a problem and how to determine whether or not its events have equal probabilities. Is that a matter of experience and intuition?
(Now, just in case the two problems turn out to indeed have different answers, I would like to ask another question. What if we are explicitly given that the balls might be distinguishable or not distinguishable with equal probabilities. What would be the answer then? 1/2, 2/3 or something in between?)
P.S. I have tried googling for my question but didn't have much luck. It still seems unlikely that I'm the first person to ask something along these lines, but I honestly just couldn't find anything. In addition to that, I'd like to say that I only just started my probability course and do not know much about the notion of probability distribution. This problem seems simple enough to be solved without using advanced theory though. Thanks.
Your question concerns the issue of how to assign prior probabilities and lies at the foundation of probability theory. Most books on probability theory simply assume a particular assignment of probability measure over the sample space (precisely $\sigma$-algebra of events). However the exercise of assigning prior probabilities lies outside the domain of probability theory and extraneous principles have to be brought in. For example the so-called principle of indifference says that if all outcomes are symmetric w.r.t. information available to you then all of them have to be assigned equal probability by virtue of this symmetry. This principle cannot be justified or derived within probability theory. There are other principles too, such as maximum entropy principle and transformation invariance. Also see Probability Theory: Logic of Science by E.T. Jaynes.
So to answer your question specifically, depending on which list of outcomes you choose, applying the principle of indifference to that list assigns equal probabilities to the outcomes in that list. Since this is a physical problem, experiments become the arbiter of which of them is correct.