The condition of $\Pr[X_1 \in S_1| X_2 \in S_2,X_3 \in S_3 ] = \Pr[X_1 \in S_1| X_3 \in S_3 ] $

Question

If $X_1,X_2$ are independent random variable, can I claim that

$$\Pr[X_1 \in S_1| X_2 \in S_2,X_3 \in S_3 ] = \Pr[X_1 \in S_1| X_3 \in S_3 ]? $$

Or else in what condition does the above equation hold?

Answer 1

Let it be that $X_1,X_2$ are independent and have non-degenerate Bernouilli distributions.

Define $X_3=X_1+X_2$.

Then: $$\Pr(X_1=1\mid X_2=1,X_3=1)=0\neq\Pr(X_1=1\mid X_3=1)$$

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A sufficient condition for the equality in your question to hold is independence of $X_1$ and $\langle X_2,X_3\rangle$.

In that case both sides equal $\Pr(X_1\in S_1)$.

This condition is not necessary (as is made clear in comments).

I took things too easily.

Answer 2

No, let $X_1 ,X_2$ are Bernoulli r.v. with values in $\{0,1\}$, i.e. independent tosses of fair coin. Let $X_3 = X_1 + X_2$. Take $S_3 = \{0,2\}$ and $S_2 = \{1\}$, $S_1 = \{1\}$. You get $$P(X_1\in S_1\mid X_2\in S_2 ,X_3\in S_3) = 1$$ while $$P(X_1\in S_1\mid X_3\in S_3) = \frac{1}{2} $$