If I toss two indistinguishable dice, what is the distribution for the sum of the two numbers I get?
Why?
probability-theory
asked 2011-11-12
user id:4890
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Let $X_1,X_2$ be the random variables denoting the outcome of your tosses. Then, $X_1$ and $X_2$ are independent and identically distributed and you want the distribution of $X_1+X_2$. The key idea you want to look up is "convolution" of the distributions of $X_1$ and $X_2$. – 2011-11-12
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Is this related to quantum mechanics? If not, why does it matter that the dice are indistinguishable? – 2011-11-12
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@joriki: Not necessarily (but maybe?), but my professor (discrete math) said it matters whether the dice are distinguishable or not, so I'd like to understand why. (He never got around to explaining it.) – 2011-11-12
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@Dinesh: I know what convolution is, but does that work for indistinguishable dice as well as distinguishable dice? – 2011-11-12
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@Mehrdad: I interpreted the word "indistinguishable" in your question to mean that the two dice are identical. Apart from this, I can't see how indistinguishability will change the solution. Distinguishable vs Indistinguishable makes a difference in Combinatorics when counting the number of ways of picking k objects out of a set of n objects. But if you have two dice tosses, you have two values whether or not the dice are indistinguishable. – 2011-11-12
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@Dinesh: Yes, it means they are identical. Which means you can't distinguish them. So the phrase "the probability of the first one coming up 3" doesn't make sense because once they are rolled, you can't distinguish the "first" one from the "second" one. I *know* this makes a difference in the outcome, but what I would like to know is what the difference is, and why (I don't know how to calculate it properly). – 2011-11-12
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So a pair of dice, on red and one green, will work differently for a colorblind person that they would for someone with normal color vision? – 2011-11-12
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@DougChatham: +1, That's a brilliant observation, and... frankly, I don't know how to answer it. I feel like it starts getting into quantum mechanics (observation affects the outcome?) but I don't know enough about it to make a call. (Having read *The Quantum Zoo*, though, I know for a fact that distinguishability at least makes a difference in quantum mechanics.) That said, I'm also hesitant to rely on common-sense reasoning like this -- I'd rather see a nice description somewhere that explains mathematically how indistinguishability does (or does not) affect the outcome. – 2011-11-12
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It is becoming increasingly obvious that the OP is not after a mathematical answer (which has been provided several times on this page) to the (trivial) question asked but rather after an answer to some worn-out paradoxes associated to indistinguishability in quantum mechanics. Two remarks are in order. First, QM applies to the behaviour of matter and energy at the subatomic level, thus, not to actual dice or coins as we know them (but maybe to *whatever*). Second, the QM question is not in the scope of this forum. – 2011-11-12
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Mehrdad, it would seem your best strategy is to print all this out, answers and all, then show that to your professor or teaching assistant and ask for comments. There is no way for you to tell us every word your professor said, hence a certain amount of mindreading in the answers. However, the reverse is readily accomplished with a printout. – 2011-11-12
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@Will Jagy: Will definitely consider doing that, thanks for the suggestion. :) – 2011-11-12
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At the same time, it would have helped if you had identified your textbook. – 2011-11-12
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@WillJagy: It's a reader, not a textbook. – 2011-11-12
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If you're interested in the quantum mechanics aspect of this, you might want to take a look at [this Wikipedia article](http://en.wikipedia.org/wiki/Identical_particles), particularly [this section](http://en.wikipedia.org/wiki/Identical_particles#Statistical_properties). Note also that what is usually described as statistical effects of indistinguishability can be less mysteriously and with fewer philosophical complications be described as statistical effects of the symmetry or anti-symmetry of states. – 2011-11-13
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@joriki: That's a great read, thanks for providing the link. (I started reading about the [Gibbs paradox](http://en.wikipedia.org/wiki/Gibbs_paradox) afterward, which is also interesting.) – 2011-11-13
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@Mehrdad: We will not delete this, as there are already two answers written by other people. – 2011-11-13