Sir,I want the proof of the probability theorem P(A and B) = P(A).P(B) where events A and B are indipendent,the number of event points of A = l ,that of B = m with n number of common event points.

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- Sep 2nd 2009, 04:18 AMspdasBasic Probability concept
Sir,I want the proof of the probability theorem P(A and B) = P(A).P(B) where events A and B are indipendent,the number of event points of A = l ,that of B = m with n number of common event points.

- Sep 7th 2009, 11:21 PMleinadprobability axioms
Hi there

Your question is an interesting one and I am a little puzzled as to why I myself have not considered this basic question! I had to look into this so I am not 100% sure of the answer but here is my best attempt;

As with all theories we often start with axioms that are simply stated or defined and accepted. All other theories derive from these.

If the sample space $\displaystyle \Omega$ has $\displaystyle N<\infty$ elements, i.e A,B C .. etc where A can contain one or more elementary outcomes then the classical axioms of probability according to Kolmogorov are;

Let A and B be any events then

$\displaystyle P(A)\geq0,$

$\displaystyle P(\Omega)=1$

$\displaystyle P(A\cup B)=P(A)+P(B)$ when $\displaystyle A\cap B=\oslash$.

The definition of conditional probability

$\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$ satisfies the above axioms.

We can state intuitively that if knowing B does not affect A, i.e $\displaystyle P(A|B)=P(A)$, then from the conditional probability definition we can state that

$\displaystyle P(A)=\frac{P(A\cap B)}{P(B)}$

which leads to the multiplication of independent events.

Thus as far as I can see the independent events that multiply comes directly from the definition of conditional probability. If we were to make another definition then this might change.