Working through some probability theory sample questions from my course notes and I'm having difficulty trying to prove this probability theorem:

Let $A_1$, $A_2$, $\ldots$, $A_n$ be events in the sample space $\Omega$ with probability measure P. Show that:

P($A_1\bigcap$ $A_2\bigcap\ldots\bigcap$ $A_n$) $\ge$ P( $A_1$) + P( $A_2$) + $\ldots$ + P( $A_n$) - (n - 1)

Any help with this would be greatly appreciated. Even a hint as to where to start. I'd like to figure it out on my own, so that I can understand it in full detail, but I am really stuck atm.

  • $\begingroup$ I think this is right - if the RHS is negative then the lower bound is trivial, but if each event has probability 1, then this gives the correct lower bound of 1. I would look up De Morgan's law and the inclusion exclusion principle. $\endgroup$ – combo Sep 5 '17 at 1:05

Prove it by induction. Prove $P(A_1 \cap A_2) \geq P(A_1) + P(A_2) - 1$. Then show $$P(B_n) \geq \sum_{k=1}^n P(A_k) - (n-1) \implies P(B_{n+1}) \geq \sum_{k=1}^{n+1} P(A_k) - n$$ where $B_n = \bigcap_{k=1}^n A_k$.

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