# Tag Info

### A and B are independent. Does P(A ∩ B|C) = P(A|C) · P(B|C) hold?

No this is not in general true, as you can see from a simple counter example: Toss two independent coins. Event $A$ is coin 1 head. $P(A)=0.5$ Event $B$ is coin 2 head. $P(B)=0.5$ Event $C$ is either ...
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### Expectation of potential outcomes formula

I assume that by "conditional independence assumption" you mean $$Y_{1i}, Y_{0i} \perp D_i | X_i.$$ The trick is simply to condition on $X_i$. Conditional on $X_i$, the propoensity score $p(X_i)$ ...
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### Independence and conditional distribution

You are considering a random sample of size $N$. Each draw $i$ performed independently of any other draw $j$. Each draw is a draw of a random vector $(Y_i,X_i)$. Here is an illustration of all the ...
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### How to reason about independence of combinations of events?

For inspiration, let's examine Venn diagrams of each set. Elements of $\mathcal{X}=A\cap B^c\cap D$ are (a) in $A;$ (b) not in $B;$ and (c) in $D$. This region is highlighted in yellow. Elements of ...
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### Are two coin flips conditionally independent if we know that the coin is biased towards heads?

The quoted section is implicitly assuming that the event $C = \{ \theta > 0.5 \}$ is sufficient to fully describe the parameter, and so it attains conditional independence of the observable coin ...
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