Say that I have sequence of iid bernoulli random variables $ (X_n)$ taking values in {0,1} with equal probability 0.5. Then I know by the weak law of large numbers that for every $\epsilon>0, \lim P({w \in \Omega: |S_n(w)- 1/2|<\epsilon})= 1$ where $S_n= X_1+...+X_n$ Would I be correct to assume that the sample space $\Omega$ would be the set of infinite sequences in {0,1}? So one possible w might be (0,1,0,1,....).? Thanks.
2026-03-25 05:06:45.1774415205
Sample space of sequence of random variables
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Yes, that is the sample space and $(0,1,0,1,\ldots)$ is an example of an outcome.
Not sure what the WLLN has to do with that, but I'll note you have a typo. It is $S_n/n$ that converges in probability to $1/2,$ not $S_n.$