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question: is P(A ∩ B) = P(A) * P(B)?

this might be an obvious question, but in our probability lectures it hasn't been mentioned by the professor and my friend and me somehow didn't manage to get an answer by looking it up on google. thanks in advance for confirming!

stanleysearles
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  • Always try examples: Suppose $A$ is the event a coin comes up Heads and $B$ is the event the coin comes up Tails. – lulu Oct 27 '22 at 18:02
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    No. $\Pr(A\cap B)$ is equal to $\Pr(A)\times \Pr(B)$ if and only if $A$ and $B$ are independent events. That is usually taken as the definition of what it means for events to be independent. Many events do happen to be independent. Many more events happen to not be independent. The more general statement is that $\Pr(A\cap B) = \Pr(A)\times \Pr(B\mid A)$ where this last term is the conditional probability of $B$ given $A$. – JMoravitz Oct 27 '22 at 18:12
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    While we're at it, since you asked this question you should also be made aware of that $\Pr(A\cup B)$ is not equal to $\Pr(A)+\Pr(B)$. That will be true if and only if $A\cap B$ is a nullset (i.e. occurs with probability zero, though not necessarily empty). The more general statement which is always true is that $\Pr(A\cup B)= \Pr(A)+\Pr(B)-\Pr(A\cap B)$ – JMoravitz Oct 27 '22 at 18:14

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