My Math Forum Probability of Infinite Union of Events

 September 26th, 2017, 10:51 PM #1 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 Probability of Infinite Union of Events Honestly have no idea where to go with this one. Since Pk is not assumed to be a constant (it could be any probability mass function), I find it difficult to take advantage of any algebraic properties. I suspect that there may be some pattern by using the inclusion-exclusion formula, but I'm not seeing anything when writing it out. I'm just really not sure how to start. Any recommendations?
 September 26th, 2017, 11:37 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,324 Thanks: 1233 well... this isn't rigorous but... $B=\bigcup_{k=1}^\infty~A_k$ $B^c = \left(\bigcup_{k=1}^\infty~A_k\right)^c = \bigcap_{k=1}^\infty~A_k^c$ $P[B]=1 \Rightarrow P[B^c]=0$ $P[B^c] = P\left[\bigcap_{k=1}^\infty~A_k^c\right] = \prod_{k=1}^\infty~(1-p_k)$ $\prod_{k=1}^\infty~(1-p_k) = 0 \Rightarrow \sum_{k=1}^\infty~\ln(1-p_k) = -\infty$ Thanks from John Travolski Last edited by romsek; September 27th, 2017 at 12:26 AM.
 September 26th, 2017, 11:39 PM #3 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 That's definitely enough. That's so clever. I wish I would be able to think of things like that. It makes so much sense considering that they're independent events. Thank you.
 September 27th, 2017, 12:27 AM #4 Senior Member     Joined: Sep 2015 From: USA Posts: 2,324 Thanks: 1233 note that you have to show the other direction as well

 Tags events, infinite, probability, union

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