So, here's the problem (1.5.12, Probability and Statistics, Degroot and Schervish 3e) :

Let $ A_1, A_2, ... $ be an arbitrary infinite sequence of events, and let $ B_1, B_2, ... $ be another infinite sequence of events defined as follows: $B_1 = A_1, B_2 = A^c_1\cap A_2, B_3 = A^c_1 \cap A^c_2 \cap A_3 ... $ Prove that

$Pr\left(\bigcup_{i=1}^{n} A_i\right) = \sum_{i=1}^{n} Pr\left(B_i\right)$ for $n = 1, 2, ... $

I'm not stuck solving it, yet; I just want to make sure I am understanding the problem. I can show that $\bigcup A_i = \bigcup B_i$ and that $B_i$ are all disjoint, which allows me to show that $ Pr(\bigcup B_i) = \sum Pr(B_i) $ for finite $n$ by induction and hence $Pr(\bigcup B_i) = Pr(\bigcup A_i)$ follows? Is there anything wrong with my argument?

Thanks for any advice, and this is homework so please don't post a full solution :)

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    $\begingroup$ I think you are correct. $\endgroup$ Jan 25, 2011 at 17:22

1 Answer 1


You are right. Informally-intutively, the B sequences selects the first true value of the A sequence. Hence, the right side (probability that at least one event A is true) is equal to the left side (sum of probabitites that the first true A event was i).


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