Let's say I have two variables $A$ and $B$ and I want to write the probability that $A=2$ conditional on $B=1$, which of the two ways would be correct:

$$P(A=2 | B=1) $$


$$ P(A=2 \cap B=1) $$

  • $\begingroup$ It's the first. The second is the probably that $A=2$ AND $B=1$. $\endgroup$ – msitt Apr 27 '17 at 15:53
  • $\begingroup$ Okay maybe I am misunderstand conditional probability then, isnt conditional probability that A=2 AND B=1? $\endgroup$ – no nein Apr 27 '17 at 15:54
  • $\begingroup$ That's the probability that $A=2$ if you know $B=1$. $\endgroup$ – msitt Apr 27 '17 at 15:55
  • $\begingroup$ So in the above example, assuming independence $P(A=2|B=1)$ is $P(A=2)$ and $P(A=2 \cap B=1)$ is $P(A) \cdot P(B)$ $\endgroup$ – no nein Apr 27 '17 at 15:58
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    $\begingroup$ By definition a "process" is a set of random variables indexed by a set of times or locations. Because you use no indexing and equate these "processes" with numbers, it doesn't appear that you are asking about processes at all. What, then, are you really asking about?? $\endgroup$ – whuber Apr 27 '17 at 16:03

The conditional probability $P(A=2 \:|\: B=1)$ is the probability that $A=2$ given that $B=1$.

The joint probability $P(A=2 \cap B=1)$ is the probability that $A=2$ and $B=1$.

In general, these two expressions are related by $$P(A=2 \cap B=1) = P(A=2 \:|\: B=1)P(B=1).$$


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