Suppose we have three discrete random variables $X_0$, $X_1$ and $X_2$. Also, suppose we know the conditional distribution of $X_1$ given $X_0+X_1=k_1$ and we also know the conditional distribution of $X_2$ given $X_0+X_2=k_2$.

Does it make sense to define the following probability:

$Pr\left[(X_1 \ge c_1 \quad \textrm{given} \quad X_0+X_1=k_1)\quad \textrm{and} \quad (X_2 \ge c_2 \quad \textrm{given} \quad X_0+X_2=k_2)\right]$?

Or we can only define the following:

$Pr(X_1 \ge c_1 \quad \textrm{and} \quad X_2 \ge c_2 \quad \textrm{given} \quad X_0+X_1=k_1 \quad \textrm{and} \quad X_0+X_2=k_2)$?


In classical probability, conditioning enforces a restriction on your event space $\Omega$, i.e. a restriction on the denominator of your probabilities (and consequently on the numerator). So it doesn't make sense to compare probabilities of two events happening on the same space under different conditioning rules. In other words, events in probability have full omnipotence of what's going on, so there's never partial information where event $X_1\geq c_1$ knows something that $X_2\geq c_2$ doesn't know and vice-versa.

Thus only the second interpretation makes sense.


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