Based on this: [Equal in distribution but unequal almost everywhere?](https://math.stackexchange.com/questions/999841)

My answer to questions like those is usually

> 
> Flip a fair coin. $1\_H$ and $1\_T$ have the same distribution but are never equal.
> 
> 
> 

I didn't consider that independence could make a difference. Okay so if $X$ and $Y$ are identically distributed, independent in a finite sample space, what's a simple example to say that they aren't a.s. equal? What about for countably infinite sample space? What about uncountable sample space but RVs still discrete?

I have a feeling I'm missing some simple example to do with flipping a coin or rolling a die.

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[If two random variables have CDFs that have the same value for all x, can we assume the random variables are equal?](https://math.stackexchange.com/questions/2746337)

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