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.
Best Answer
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!