If A and B are two dependent events then:
$P(A\cap B) = P(A).P(B|A) $
How can we prove this logically(NOTE: not analytically, i.e. not just using algebraic equations but with reasoning)?
Proving probability of A and B for events
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probability
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0How do you define the conditional probability $P(A\,|\,B)$? – 2017-01-08
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0@martini, sorry, a simple typo – 2017-01-08
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0That is the very definition of $P(B\mid A)$, there is nothing to prove logically, so it is quite unclear to me what you are asking – 2017-01-08
1 Answers
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I think it makes more sense if you consider sets$A$ and $B $ on a Venn diagram. The intersection $A \cap B$ contains those elements that are in set $A $ and in set $B $
The ratio $$\frac{(A \cap B)}{ (A)}$$ is really just the probability of getting $B $ assuming you are already in $A $.
Dividing through by an the total number of elements yields the form you are familiar with.