Comma vs pipe/vertical line in notation for conditional probability

Question

What is the difference between the following expressions:

$$P(X_1 < X_2 \mid \min(X_1, X_2) = t) \qquad \text{and}\qquad P(X_1 < X_2, \min(X_1, X_2) = t)$$

For context, I am trying to understand what it means to say that $$P\left(X_1 < X_2 \mid \min(X_1, X_2) = t\right) = \frac{P\left(X_1 < X_2, \min(X_1, X_2) = t\right)}{P(\min(X_1, X_2)=t)}$$

Answer 1

You can read the symbols $\mid$ and $,$ as follows:

1. $"\mid"=$ given that. So the LHS can be read as: Probability that $X_1$ is less than $X_2$ given that the minimum of $X_1, X_2$ equals $t$.
2. $","=$ and. So the RHS (numerator) can be read as: Probability that $X_1$ is less than $X_2$ and that the minimum of $X_1, X_2$ equals $t$.

In the first case, it is given/ known/ has already happened that the minimum of $X_1$ and $X_2$ is equal to $t$ (and you know it), whereas in the second case you want to find the probability that $X_1<X_2$ and simultaneously that the minimum of $X_1$ and $X_2$ equals $t$.

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In this specific case the RHS (numerator) probability can be simplified to $$P(X_1<X_2, \min\{X_1,X_2\}=t)=P(X_1=t, X_2\ge t)$$ However in most cases the LHS (conditional probability) is easier to calculate.