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Suppose a coin is flipped infinitely many times and we take as our sample space $S$ all possible infinite length sequences of heads and tails. For $n\geq1$, let $A_n$ be the event that in the first $n$ flips the proportion of heads is at least $1/2$.

(a) Express in terms of the events $A_n$ using set operations the event $A$, which is defined as the set of all sequences in $S$ with the property that for some $k$, the proportion of heads in the first $n$ flips is at least $1/2$ for all $n\geq k$.

Could someone help me parse this out a bit. I don't have trouble calculating actual probabilities but the set operation language and notation is a bit confusing for me. Any help would be appreciated.

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  • $\begingroup$ Hint: Assuming the $A_n$ are as described, given some integer $k$, what is $\bigcap_{n \ge k} A_n$? $\endgroup$ – BrianO Oct 19 '15 at 6:12
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Let's start from the end. We want $A_n$ to occur for all $n\ge k$, i.e., $A_k$ and $A_{k+1}$ and ... . "And" translates to intersection, so this event is $A_k\cap A_{k+1}\cap \ldots$ or more succinctly $B_k:=\bigcap _{n\ge k} A_n$. This again shall happen "for some $k$", so for $k=1$ or $k=2$ or $k=3$ or ... . "Or" translates to union, so this event is $B_1\cup B_2\cup\ldots$, or in short $\bigcup_{k\in\Bbb N}B_k$. In summary, $$\bigcup_{k\in\Bbb N}\bigcap_{n\ge k}A_n.$$

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