2 6-sided die are rolled. What is the probability that their faces are different?

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I thought it was:

$$ \frac{1}{6}\frac{5}{6} = \frac{5}{36} $$

since the first die has one number out of $6$, and the second die has to be the other $5$.

But I'm not exactly sure if that's right.

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No matter the outcome of the first die, the second die has a $1$ in $6$ chance of matching that. So, the probability of being different is $5$ out of $6$.