# Conditional probability between parameter and operator in quantum mechanics?

## Background

So I came across a question on conditional probability in quantum mechanics: There's an interesting comment which tells why this does not work for "the non-commutative case".

I was wondering, however, since there are more than operators in quantum mechanics one could ask about their relation. For example, there is time which is a parameter. It seems straightforward to compute the conditional probability of an outcome given the time was say $$t$$ by (for example):

$$P( A|T_1) = |\langle x_A, t_1 | \psi, t_1 \rangle|^2$$

where $$A$$ denotes the event of say measuring the position at a $$x = x_a$$, $$T_1$$ represents the time being say $$t_1$$ and let pre-measurement state be $$\psi$$. But what if one swaps things as:

$$P(T_1|A) = \frac{| \langle x_a, t_1| \psi, t_1\rangle |^2}{\int_{t_{A-}}^{t_{A+}} | \langle x_a, t| \psi, t\rangle |^2 dt}$$

This something I figured out here: Physics SE: Conditional probability between parameter and operator in quantum mechanics

## Question

How does this change if we update the question to multiple measurements. For example what is the probability of $$P(A|T_1)$$ and $$P(B|T_2)$$ (with $$T_2 > T_1$$ as the time evolution of the wave-function to see $$B$$ at $$T_2$$ depends on the outcome A) - here $$A$$ and $$B$$ can be different observable's outcomes? How is this related to probability of the times being $$T_1$$ and $$T_2$$ when $$A$$ and $$B$$ are observed?