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So I've been working in probability regarding dice rolls. I came across this problem:

If you roll 2 dice, what is the probability the first die is a 6 given that you rolled an 8?

This is clearly a conditional probability where E = {the event you roll an 8} and S = {event you roll a 6}. So the P(S) = 1/6. Then P(E) = 5/36. P(E) is calculated using the fact that there 36 possible permutations of rolls and 5 ways to rolls an 8 ({(2,6), (6,2), (3,5), (5,3), (4,4)}).

I'm just curious why (4,4) isn't counted twice? What is the logic behind this?

I'm assuming it has something to do with it just not being distinguishable? As in if I say dice 1 = 5 and dice 2 = 3 versus dice 1 = 3 and dice 2 = 5, there's a distinction, but saying dice 1 = 4 and dice 2 = 4 is not distinct form dice = 4 and dice 2 = 4, but this isn't satisfactory explanation for me.

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  • $\begingroup$ Each of the events, the five ways to get an $8$, have the same probability, namely $\frac 1{36}$. No reason to count any of them twice. $\endgroup$ – lulu May 25 '19 at 14:29
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Here's another way I like to think about it. Throwing two die at the same time is equivalent to throwing them one by one. Now look at the question again.

Are $(2,6)$ and $(6,2)$ different? Yes, indeed!

Now what about $(4,4)$? Nope!

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The only way to get $(4,4)$ is if both dice show $4.$ By contrast, there are two distinct (if not distinguishable) ways for the dice to show a $2$ and a $6,$ which is why we count both $(2,6)$ and $(6,2).$

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Pretend that the two die you are rolling have different colors - say red and blue. If red shows 2 and blue shows 6, that's different from if it's the other way around. But if red shows 4 and blue shows 4, then that's the same outcome as if blue shows 4 and red shows 4.

The sample space has 36 pairs, of which only one of them is (red=4, blue=4). Two of the 36 outcomes are (red=2, blue=6) and (red=6,blue=2). There are 6 ways to choose the number for red and 6 ways to choose the number of blue, for a total of 36 outcomes. Think of the set of positions in a 6 by 6 matrix as the set of all outcomes. If (4,4) was counted twice, then you would have 37 or more positions.

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