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- Let me borrow your cleverness, please?

I have six events with known probabilities ##p_1, ..., p_6##. Find the probability of two or more of these events occurring together? I can't think of a clever way to calculate this without using the problematic "or" is addition rule, but using that rule I get the required probability is

P(2 or more events) ##= 1-\prod_{k=1}^{6}\left( 1-p_{k}\right) - \sum_{j=1}^{6}p_{j} \prod_{l\neq j}^{6}\left( 1-p_{l}\right)##

P(2 or more events) ##= 1-\prod_{k=1}^{6}\left( 1-p_{k}\right) - \sum_{j=1}^{6}p_{j} \prod_{l\neq j}^{6}\left( 1-p_{l}\right)##