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I have been using several books and I came across different notations that I am assuming they mean the same thing, but I am not 100% sure.

Are these probabilities the same thing?

$P(AB) \equiv P(A,B) \equiv P(A \land B) \equiv P(A \cap B) \equiv P(A \times B)$?

Edited: added $P(A \times B)$

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  • $\begingroup$ $P(AB)$ looks more like $P(A \times B)$. $\endgroup$ – Tim Feb 5 at 20:53
  • $\begingroup$ So, $P(A,B) \not \equiv P(A \times B)$? $\endgroup$ – Fred Guth Feb 5 at 21:09
  • $\begingroup$ I think the notation just originates from early developments in modern logic more precisely Boole and Boolean Algebra. In this tradition it is standard to associate mutiplication with logical AND and addition with logical inclusive or. Then since $AB$ and $A\times B$ are standard for multiplication they are were used for AND (although probably not so much any longer). You could just think of multiplying indicator functions getting 1 if and only if both indication $I[\omega \in A]$ and $I[\omega \in B]$ are equal to 1 hence the same as logical AND also denoted $\wedge$. $\endgroup$ – Stop Closing Questions Fast Feb 5 at 22:26
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  • $P(A, B)$ is a pretty standard notation for joint probability two events $A$, $B$, both being true,
  • $P(A \cap B)$ means the same, but it is using set theory notation for intersection, in probability theory it is just as a notation used for saying $A$ and $B$,
  • $P(A \land B)$ is the same, but is using a notation borrowed from logic for conjunction,
  • $P(AB)$ or $P(A \times B)$ is a confusing choice of notation; as noticed in the comment if you multiply binary numbers then $AB = A \land B$, so it would be equivalent.

    However, if we start talking about random variables, then the notation $p(x, y) = P(X = x, Y = y)$ still has the same sense, while $p(xy)$ is not the same thing, since this is a distribution of a product of two random variables, that is not the same as their joint distribution (both formally, since $p(xy)$ is a function of single variable, while $p(x,y)$ is a function of two variables, and in terms of the probabilities returned by it). To give an example, say that you ask what is $P(X=5, Y=3)$, since $5\times 3 = 15$, then $XY = 15$ would be true both if $x=3$ and $y=5$, and if $x=5$ and $y=3$. This would be the case, informally speaking, if there was some kind of equivalence between $X$ and $Y$, and they were independent of each other.

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$P(AB)$ is not a standard notation, $AB$ can be the name of the event, or as in your example, each letter can denote different events. But, $P(A,B)$, which maybe used for shorthand purposes, and others most probably mean the same thing.

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  • $\begingroup$ In your opinion, $P(AB)$ is a bad notation because it may lead to confusion (you bet it), ok. I added $P(A \times B)$, do you think it is also the same? $\endgroup$ – Fred Guth Feb 5 at 21:10
  • $\begingroup$ I don't recall seeing that notation for referring probabilities. It's like a cartesian product. Not saying that it is invalid, but probably rare. All books and lecture notes use their own notation, although they have a lot in common. $\endgroup$ – gunes Feb 5 at 21:20
  • $\begingroup$ Why $P(A,B)$ may be used for shorthand? What do you mean? $\endgroup$ – Fred Guth Feb 6 at 0:24

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