Strange sum notation

Question

I have the following sum and I would like to generalize the notation but I don't see how to choose i index...

$$ -2P(A_{1} \cap A_{2}) - 2P(A_{1} \cap A_{3}) - 2P(A_{2} \cap A_{3}) = -2\sum_{i=something}^{something} P(A_{i}\cap A_{j}) $$

thanks ;)

Answer 1

It depends on what you want to achieve. If you want to sum all possibile combinations of two events, then using a double summation will do the trick:

$$\sum_{i,j\\ i\neq j}P(A_i\cap A_j)=\sum_{i}\sum_{j,\ j\neq i}P(A_i\cap A_j)$$