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This seems like a trivial question but somehow I have not been able to locate a source that will state that $$P(A\cap B) = P(A,B)$$. Of course it is implied when conditional probabilities are stated as $$P(A|B)=\frac{P(A,B)}{P(B)}$$ and $$P(A|B)=\frac{P(A\cap B)}{P(B)}$$ I am missing anything? Thanks for advice and apologies if necessary.

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  • $\begingroup$ Here's an academic source: sites.nicholas.duke.edu/statsreview/probability/jmc $\endgroup$
    – whuber
    Oct 3, 2016 at 18:45
  • $\begingroup$ Thanks very much. Very helpful. Quite obvious but somehow difficult to find in some textbooks. $\endgroup$
    – LDBerriz
    Oct 3, 2016 at 18:49
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    $\begingroup$ Yes, I can guess why: an author would adopt one notation or the other, but probably not use both. In my (quick) search I also found that the language of "joint" probability focuses on particular kinds of intersections rather than general intersections: joint probability is a concept attached to products of probability spaces whereas intersections make sense in all cases. $\endgroup$
    – whuber
    Oct 3, 2016 at 18:52
  • $\begingroup$ That is exactly where my doubts came from as I am trying to explain sets and probabilities to an audience that is not supposed to know any thing about probability theory so I thought that combining the explanation in a table of $P(A,B)$ with a Venn diagram that showed $P(\cap B)$ would be easier to understand it. $\endgroup$
    – LDBerriz
    Oct 3, 2016 at 18:59
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    $\begingroup$ I always just saw these as two ways of notating the probability of an intersection, but I always preferred being explicit about the intersection with $\cap$. $\endgroup$
    – dsaxton
    Oct 3, 2016 at 21:32

2 Answers 2

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Short answer: yes

Long answer: probability is just a measure of the likelihood of some set of events happening (e.g. a coin flip landing heads is the event, and the probability of this event for a fair coin is 0.5). So if you're looking at 2 sets that are exactly the same, namely A intersect B or A,B (although I find the latter notation a bit ambiguous), then their probabilities have to be the same. In general, it's a lot easier to think about events and how they relate, then using that to construct your probabilities.

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Yes, commas generally are used to denote intersection even by those who are otherwise very careful to avoid the possibility of their writings being misinterpreted. The most common usage is $P(X\leq x, Y \leq y)$ for the more prolix and correct $P\big ( (X \leq x)\cap (Y \leq y)\big)$ to denote the value of the joint CDF of random variables $X$ and $Y$.

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  • $\begingroup$ My impression is that the use of the comma is when one is using tables in which case seems intuitive to express the joint probability of row,column intersection as $P(r,c)$ whereas when dealing with diagrams $P(A \cap B)$ seems more appropriate. $\endgroup$
    – LDBerriz
    Oct 4, 2016 at 0:41

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