Please can someone explain why the order of events with conditional probability does not matter?
p(A and B) = p(A | B) x p(B) = p(B | A) x p(A)
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7$\begingroup$ Would it help to observe that the event "A and B" is identical to the event "B and A"? $\endgroup$– whuber ♦Commented Mar 6, 2019 at 13:05
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$\begingroup$ Why should it matter? $\endgroup$– TimCommented Mar 6, 2019 at 13:35
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$\begingroup$ I was thinking about time and causality and found it confusing. I really needed someone to explain to me that they are unimportant to conditional probability. $\endgroup$– MichaelAndroidNewbieCommented Mar 6, 2019 at 16:02
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1 Answer
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I think you are asking a pretty good question despite seeming innocent at first. My answer would be that, from a matehmatical point of view, "B given A" or "B/A" does not necesarelly mean that A happened before B.
A could have happened before, after or at the same time as B, and causal relationships between A and B could go either way, but what P(B/A) really means is "OK: We know A happened/is happening/will happen, now what is the chance that B happenned/is happening/will happen as well?"
In these terms, keep in mind that we don't know everything that happenned in the past, neither do we ignore everything that will happen in the future.
Silly example: I am going to roll two dice. In experminet A I throw one of them first, getting a 4. What is the probability of the sum of both dice being 9? In case B, I have not rolled any of the dice yet, but I know the second one is rigged and it will always be a 4. What is the probability of the sum of both dice being 9 now? Of course the same as before!
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4$\begingroup$ Although you're right, I am concerned this answer might make the issue even more confusing because it introduces considerations of causality without making it very clear that the concept and definition of conditional probability have nothing whatsoever to do with time or causality. $\endgroup$– whuber ♦Commented Mar 6, 2019 at 15:57
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$\begingroup$ It was time and causality that confused me. I really needed someone to explain that they are unimportant and why. I feel much more comfortable with conditional probability and Bayes theorem after this answer. Thank you! $\endgroup$ Commented Mar 6, 2019 at 16:04
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2$\begingroup$ @MichaelAndroidNewbie I’d recommend Judea Pearl’s latest book “Book of Why” (amazon.com/Book-Why-Science-Cause-Effect/dp/046509760X) to better understand why observational calculations (e.g P(X|Y) ) have no causal interpretation without hypothesizing a causal model. $\endgroup$– mattCommented Mar 6, 2019 at 16:59
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$\begingroup$ @whuber Thank you for your comment. I think your concern is right, however I think my answer is precisely erasing that confusion! $\endgroup$– DavidCommented Mar 7, 2019 at 7:31