I am trying to prove this question, but I am not sure if what I did is correct..

q: Suppose that A1, A2, ..., An are events in a random experiment whose intersection has positive probability. Prove the multiplication rule of probability.

$\displaystyle P({A_1} \cap {A_2} \cap....{A_n})$ $\displaystyle = P(A_1) P({A_2} | {A_1}) P({A_3} | {A_1} \cap A2) ...... P({A_n} | {A_1} {A_2}......{A_{n-1})$

This is what I have done:

$\displaystyle P(A_1) = P(A_1)$

$\displaystyle P({A_2}\cap{A_1}) = P({A_2}|{A_1}) P({A_1})$ [by the definition of conditional prob]

$\displaystyle P[{A_3} \cap ({A_2}\cap{A_1})]$ $\displaystyle = P({A_3} |{A_2}\cap{A_1}) P({A_2}\cap{A_1})$ $\displaystyle =P({A_3} |{A_2}\cap{A_1}) P({A_2}|{A_1}) P({A_1}) $

now assuming this is true for all n, we have

$\displaystyle P[{A_{n+1}} \cap ({A_1} \cap ...\cap{A_n})]$ $\displaystyle = P({A_1})P({A_2}|{A_1}) P({A_3} |{A_2}\cap{A_1}).......P({A_{n+1}}|{A_1} \cap.......\cap{A_n})$

thus it is true for all n..

Is it path I have followed correct? Any inputs will be appreciated!

Thanks