Please explain the question posted in the file attached.

CB

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- Jun 19th 2007, 07:51 PMcomputer-botProblem with equation of independent events
Please explain the question posted in the file attached.

CB - Jun 19th 2007, 08:42 PMCaptainBlack
$\displaystyle A \cap B$ is the set of simple events which are in both $\displaystyle A$ and $\displaystyle B$, and $\displaystyle |A \cap B|$ is the number of elements in this set.

It is the number of ways the both $\displaystyle A$ and $\displaystyle B$ can simultaneously occur.

RonL - Jun 20th 2007, 05:24 AMcomputer-bot
CaptainBlack I know what does A intersection B means but what I don't understand is that what does this statement |A and B|/|B| has to do with the occurence of two events. I mean the author said that "The probability that A occurs is P(A) = |A|/6 = 3/6 = 1/2, while presuming B occurs, the probability that A occurs is

**|A and B|/|B|.**What I am trying to understand is that how did the author form this equation for this situation. I hope my question is clear. - Jun 20th 2007, 07:08 AMPlato
The probability of A happening given that B has occurred in symbols is

$\displaystyle P(A|B) = \frac{{P\left( {A \cap B} \right)}}{{P(B)}}$.

That explains the intersection in your question.

Thus if A and B are independent that means that

$\displaystyle P(A|B) = \frac{{P\left( {A \cap B} \right)}}{{P(B)}} = P(A)\quad \Rightarrow \quad P\left( {A \cap B} \right) = P(A)P(B)$.