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Let's consider the General Multiplication Rule in Probability, defined as:
$P(A\cap B)=P(A)P(B|A)$.
I don't understand why this formula commutes, that is:
$P(A\cap B)=P(B\cap A)$
to me, according with the general formula these 2 probabilities can be different, what do you think ?
2 Answers
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The sets $A \cap B$ and $B \cap A$ are the same.
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$A \cap B$ and $B \cap A$ both mean both $A$ and $B$ happen, leading to equality of the probability.
This leads to Bayes' formula for reversing the order of conditioning. $$ P(B|A)=\frac{P(A|B)P(B)}{P(A)} $$
