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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 inclusionexclusion 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... $B=\bigcup_{k=1}^\infty~A_k$ $B^c = \left(\bigcup_{k=1}^\infty~A_k\right)^c = \bigcap_{k=1}^\infty~A_k^c$ $P[B]=1 \Rightarrow P[B^c]=0$ $P[B^c] = P\left[\bigcap_{k=1}^\infty~A_k^c\right] = \prod_{k=1}^\infty~(1p_k)$ $\prod_{k=1}^\infty~(1p_k) = 0 \Rightarrow \sum_{k=1}^\infty~\ln(1p_k) = \infty$ 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


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events, infinite, probability, union 
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