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I'm trying to get a better intuition for (infinite) independent sequences of events and how the union/intersection affects this.

1) If an event $B$ is independent with $A_k$, is it independent with:

  • $\cup_{i \geq 1} A_k$?
  • $\cap_{i \geq 1} A_k$?

2) Does additional information that the sequence is decreasing ($A_{k+1} \subset A_{k}$) or increasing affect the answers to the above question?

3) Is there any intuitive way to see immediately that unions, intersections, and other manipulations of these events result in independence? I often feel like I'm trying to brute force my way through algebra (using continuity of probability and other techniques in an introductory graduate probability theory class), but not quite understanding fully.

In general I don't think I have a good framework for thinking about independence beyond the basic definition taught ($P(A \cap B) = P(A)P(B)$, information about one doesn't affect the other). If anyone has some good resources/exercises similar to the ones above to feel comfortable with the concept and unintuitive cases (like how pairwise independence doesn't imply mutual independence), I would be grateful and happy to learn.

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(1) Here is an example with finite collection of events. Take $n$ independent fair coin tosses and let $A_k=\{\text{heads at the $k$th toss}\}$. Let $B=\{\text{even number of heads in $n$ tosses}\}$. Then $B$ is independent of $A_k$ because $$ \mathbb{P}\{B\mid A_k\}=\frac{1}{2^{n-1}}\sum_{i=0}^{\lfloor{(n-1)/2}\rfloor}\binom{n-1}{2i+1} \\ =\mathbb{P}\{B\}=\frac{1}{2^{n}}\sum_{i=0}^{\lfloor{n/2}\rfloor}\binom{n}{2i}=\frac{1}{2} $$

But $B$ is not independent of $C_1:=\bigcap_{k}A_k$ because $\mathbb{P}\{B\mid C_1\}=1$ or $0$. Also $C_2:=\bigcup_k A_k$ is not independent of $B$.


(2) If $A_k\subseteq A_{k+1}$ then $\{B\cap A_k\}\subseteq \{B \cap A_{k+1}\}$ and $$ \mathbb{P}\left\{B\cap \bigcup A_k\right\}=\mathbb{P}\left\{\bigcup\{B\cap A_k\}\right\}=\lim_{k\to \infty}\mathbb{P}\{B\cap A_k\} \\ =P\{B\}\times \lim_{k\to \infty} \mathbb{P}\{A_k\}=\mathbb{P}\{B\} \times \mathbb{P}\left\{\bigcup A_k\right\}. $$

The same holds for intersections and decreasing sequences.

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