0

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 ${\displaystyle P(A|B)={\frac {P(A\cap B)}{P(B)}}}$?

gung - Reinstate Monica
  • 132,789
  • 81
  • 357
  • 650
Tin
  • 101
  • 1
  • 6
    $$P(A\mid B) = \frac{P(A\cap B)}{P(B)}$$ 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 $P(A\mid B) = \frac{P(A\cap B)}{P(B)}$. – Dilip Sarwate Nov 10 '16 at 16:57
  • 1
    My answer [here](http://stats.stackexchange.com/a/239042/127790) has some pictures that might help. – GeoMatt22 Nov 10 '16 at 18:49

1 Answers1

5

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
user190080
  • 200
  • 3
  • 14