# Throwing a die - what is a probability space

I read a probability book that says:

Let us roll 2 symmetrical cubes. Determine the probability of an event that a different number on both cubes has fallen?

The solution start with writing of $$\Omega$$. $$\Omega = \{ (i,j):i,j=1,2,3,4,5,6 \}$$ My question is why Omega isn't filled with set of $$\{i,j\}$$ because if I have 2 the same dice why I count event (3,4) and (4,3)?

• because the probability of (4,4) is 1/36 and probability of (3,4) is 1/18 even if the dice are the same, indistingushable – Kostas Nov 23 '19 at 21:08
• – Kostas Nov 23 '19 at 21:09

If the dice are indistinguishable, then the outcomes are: $$(1,1), (1,2), \ldots , (6,6)$$ and of course these are not equally likely, as governed by simply binomial distribution. For instance, it is twice as likely you get $$(1,2)$$ than $$(1,1)$$.