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The conditional probability of event $A$ given event $B$ is defined as:

$ P(A|B) = \frac{P(A \cap B)}{P(B)} $

In this definition, is there any constraint that events $A$ and $B$ should be associated with the same random experiment? Or can they be events associated with different random experiments?

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  • $\begingroup$ It is not about experiments here. It can be any two events A and B. So 'yes' to your second question. 'no' to the first. $\endgroup$
    – Laksan Nathan
    Commented Sep 8, 2017 at 18:10
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    $\begingroup$ I wonder what you mean by "random experiment." Note that for "$A\cap B$" to make sense, $A$ and $B$ must be subsets of a common set. @Inathan Recall that events are defined to be measurable sets in a probability space. I believe the question might concern whether $A$ and $B$ must be events in the same probability space. $\endgroup$
    – whuber ♦
    Commented Sep 8, 2017 at 18:11
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    $\begingroup$ @whuber: yes but $A$ might be heads on the flip of a coin while $B$ might be a total of $7$ on the roll of two dice. The probability space would then have to contain both the coin flip and the dice roll $\endgroup$
    – Henry
    Commented Sep 8, 2017 at 18:14
  • $\begingroup$ @Henry thanks for adding this. exactly the idea I've had in mind $\endgroup$
    – Laksan Nathan
    Commented Sep 8, 2017 at 18:15
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    $\begingroup$ Yes, I think you have it exactly right. The very act of writing a conditional probability implies $A$ and $B$ are events in the same space. Although you can always put events from different spaces $A\subset\Omega_A$ and $B\subset\Omega_B$ into a common space (simply form their product $\Omega_A\times\Omega_B$ and identify $A$ with $A\times\Omega_B$ and $B$ with $\Omega_A\times B$), this isn't worth much, because then $\Pr(A\mid B) = \Pr(A)$. $\endgroup$
    – whuber ♦
    Commented Sep 8, 2017 at 18:44

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Answered in comments, summarized here: To form the conditional probability $P(A\mid B)$ then specifically (by the defining formula) $A \cap B$ must have a meaning, and that requires that $A$ and $B$ both are subsets of some larger set containing both. So, the events $A$ and $B$ both must be defined on the same probability space.

But say we have two "experiments" (in the colloquial sense), like throwing a coin and then rolling a dice, (event $A$ related to coin, event $B$ related to dice). Then, there is a modeling choice between defining one probability space for the coin toss, and another for the dice roll, or one probability space containing both. But, if you choose the first option, there is no way to define/calculate the conditional probability $P(A \mid B)$. For that to be defined you must include coin and dice in the same probability space.

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