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I'm taking an intro to statistics class and currently we are covering conditional probability. The chapter explained what it is quite well, but it didn't explain how the formula for it was derived, i.e it didn't explain explain how

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

was derived.

I would greatly appreciate if someone could show the reasoning as well as derivation of this formula.

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The formula for conditional probability is actually taken as somewhat of an axiom in probability and isn't really derived. It makes sense if you think of conditioning as reducing the sample space of an experiment, and remember that probabilities are always measured in relation to the total sample space. In particular they're always proportions of the total sample space.

When we reduce the sample space to $B$ then the only way for $A$ to occur is if $B$ does as well, hence the $P(A \cap B)$. And the size of this event with respect to the new sample space is $P(A \cap B) / P(B)$, or the fraction of $B$ taken up by $A \cap B$.

It can also help to imagine a Venn diagram when thinking about conditional probability. Whenever you condition on an event you restrict yourself to being in one of the circles, and all conditional probabilities are measured as fractions of the area of that circle.

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  • $\begingroup$ Interesting that it is an axiom and cannot be derived. This should be like proving NP=P. I personally appreicate that this student is thinking on first principles and not accepting non-sense. $\endgroup$ – user46925 Feb 29 '16 at 2:42

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