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1st part of my question:

I have that $$P\left(\bigcup_{i=1}^{{2^n-n}}E_i\right)$$ , how would I write it out using the inclusion-exclusion principle? I know it starts off: $$\sum_{i=1}^{2^n-n} P(E_i)+...$$ But after that Im not sure what goes next.

2nd part --- I also read somewhere that (by subadditivity), $P\left(\bigcup_{i=1}^{{2^n-n}}E_i\right) \le \sum_{i=1}^{2^n-n} P(E_i)$, but why is that the case? I dont understand how it by subadditivity the above inequality comes about.

Thanks.

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    Is there any reason for considering $2^n-n$ events $E_i$ rather than $n$ events which would simplify the notation? That is, do the $E_i$ have more specific meaning that you are not revealing to us? For example, ignoring a possible typographical error in the upper limit, you _could_ actually be wanting to find the probability that two or more of $n$ events have occurred.2012-01-23

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