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To my understanding, $P(A|B)$ is the probability of A given that B has already occur (or we know will occur): $P(A|B) = P(A \cap B) / P(B)$

What confuses me a lot is how to update the probability of A given that I know the value of $P(A|B)$ and also that B might occur with a probability $P_B$ but has not happened yet.

For example:

Let's say Bob enters a store to pick a movie. Initially, the chances of Bob choosing a movie are $\{\beta_1,...\beta_n\}$ for the all the $\{m_1,...m_n\}$ movies in the store. Then, he see his friend Anne also choosing a movie. Since Bob wants to see a movie with Anne, the chances that Bob accepts to see the move Anne chooses are known to be higher ($P(A_m|B_m)$ isn't?). However if Anne chooses a very bad movie, Bob will still take a different one. The chances of Anne choosing a movie from the store are $\{\alpha_1,...\alpha_n\}$ for the all the $\{m_1,...m_n\}$ movies in the store.

The question is what are the chances of Bob picking a given movie $m$ after seeing Anne but before knowing which movie Anne will choose?

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    More correctly, $P(A\mid B)$ is "The probability that $A$ will occur, given that $B$ will occur". Mathematics has no care for the order of events. – 2017-01-19
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    ok, I will update the question – 2017-01-19
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    El Marce, a deleted (-4)votes question is worse than a closed -4vote question. Deleting such a poor question weighs much heavily on you, in terms of maintaining the privilege to ask questions, than a question with four downvotes. So don't comfort yourself in thinking the question and the deletion of the question go away upon deletion. I say this as an FYI. If you could care less, that's fine. At least I know and you know, that evading consequences (e.g. deleting a very bad question) doesn't work here. – 2017-07-13
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    get a life amwhy, I'm sure your behavior of searching for all my questions in the site and downvoting them is also frowned upon. – 2017-07-13

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the fact that you (the mathematician) know the probability for event $B$ should not affect the probability of event $A$. $P(B)$ is a certain real, whether or not this value has been computed has no effect on $P(A)$.

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    Thank you @Jorge. My intuition tells me that with more information the probabilities should change, so I'm probably modeling my problem in the wrong way. – 2017-01-19