The definition of mutual independence of events \(\displaystyle A_{1},..A_{n}\), state that that the events should satisfy equations like \(\displaystyle p(A_{i}A_{j})=P(A_{i})P(P_{j}),P(A_{i}A_{j}A_{k})=P(A_{i})P(A_{j})P(A_{k}), .....P(A_{1}...A_{n})=P(A_{1})P(A_{2})....P(A_{n})\)

These comes down to a total of \(\displaystyle 2^n-n-1\) equations.I understand this much.

My book says that it is readily seen that this definition can be written as a set of \(\displaystyle 2^n\) equations, obtained from the last equation on replacing an arbitrary number of events \(\displaystyle A_{j }\) by their complements \(\displaystyle A'_{j}\), using the fact that if \(\displaystyle A\) and \(\displaystyle B\) are independent then \(\displaystyle P(AB')=P(A)P(B')\)

I cant understand what it means or how they arrive at \(\displaystyle 2^n\) equations, or even how it gives 4 equations for 2 events, using the complements.

Please help me.

These comes down to a total of \(\displaystyle 2^n-n-1\) equations.I understand this much.

My book says that it is readily seen that this definition can be written as a set of \(\displaystyle 2^n\) equations, obtained from the last equation on replacing an arbitrary number of events \(\displaystyle A_{j }\) by their complements \(\displaystyle A'_{j}\), using the fact that if \(\displaystyle A\) and \(\displaystyle B\) are independent then \(\displaystyle P(AB')=P(A)P(B')\)

I cant understand what it means or how they arrive at \(\displaystyle 2^n\) equations, or even how it gives 4 equations for 2 events, using the complements.

Please help me.

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