Why does not the order of occurrence of events make 2 cases?
- case when $A$ occurs first then $B$
- case when $B$ occurs first then $A$
Hence, $P(A \cap B) = \text{ case }1 + \text{ case }2 = P(A) \cdot P(B \mid A) + P(B) \cdot P(A \mid B)$.
Why conditional probability $P(A \cap B) =P(A \mid B) \cdot P(B)$? It should be $=P(A \mid B) \cdot P(B)+P(B \mid A) \cdot P(A)$
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probability
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2Don't think of conditional probability as an order, think of $P(A|B)$ as the probability $A$ occurs when $B$ occurs. – 2017-02-26
2 Answers
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It is because the definition of $P(A\,\vert\, B)$ is that it is the probability of the event ‘$A$ and $B$’ relative to the probability of the event $B$: $$P(A\,\vert\, B)=\frac{P(A\cap B)}{P(B)}.$$ Your second formula is $\;2P(A\cap B)$.
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Take a look at our daily language. Lets consider the probability of taking an umbrella and presence of a rain $P(U,R)$.
The rain occuring with prob. $P(R)$ and after we look at the window to know weither it rains or not we take a decision $P(U|R)$ if to take an umbrella. So, the actual probability is $P(U,R) = P(R)P(U|R)$. There should not be any other term.