I've been doing some learning for Native Bayes classification. I came across this formula, but I'm having trouble remembering it because I don't know how to get this formula. Can anyone explain how to get ${\displaystyle P(A|B)={\frac {P(A\cap B)}{P(B)}}}$?

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    $\begingroup$ $$P(A\mid B) = \frac{P(A\cap B)}{P(B)}$$ is the universally accepted definition of what $P(A\mid B)$ means, and there is no way to "get this formula" unless you have some alternative definition of the meaning of $P(A\mid B)$ from which we could arrive at $P(A\mid B) = \frac{P(A\cap B)}{P(B)}$. $\endgroup$ Nov 10 '16 at 16:57
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    $\begingroup$ My answer here has some pictures that might help. $\endgroup$
    – GeoMatt22
    Nov 10 '16 at 18:49

As already pointed out in the comment, the statement $$ {\displaystyle P(A|B):={\frac {P(A\cap B)}{P(B)}}} $$ is the very definition of the conditional probability. Check for example here on wikipedia for an overview.

If you have trouble to remember the definition,

  • maybe try to imagine how you would try to literally condition on something (event).
  • how would you write such an expression, if you would condition on the whole event space $\Omega$ - what would you expect as an outcome
  • and for the peace of mind, show that ${\displaystyle P(\cdot|B):={\frac {P(\cdot\cap B)}{P(B)}}}$ is indeed a probability measure itself

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