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August 31st, 2017, 09:22 PM   #1
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Exclamation Probability of Union of Events

Without going through the effort of actually manually expanding each term through union and intersection laws (there's a lot of distributing that would have to be done), I'm not quite sure how to solve this problem:

Suppose P(Ai) = 1/(3 + i) for ¡ = 1, 2, 3, 4. Find an upper bound for
P(A1 U A2 U A3 U A4).

There must be an easier approach using some applicable theorem that fails to come to mind at the moment. Any advice?

Also, I'm not sure what the question is referring to with the term "upper bound." How does that value compare to the probability of the union of the four events?
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August 31st, 2017, 09:34 PM   #2
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Quote:
Originally Posted by John Travolski View Post
Without going through the effort of actually manually expanding each term through union and intersection laws (there's a lot of distributing that would have to be done), I'm not quite sure how to solve this problem:

Suppose P(Ai) = 1/(3 + i) for ¡ = 1, 2, 3, 4. Find an upper bound for
P(A1 U A2 U A3 U A4).

There must be an easier approach using some applicable theorem that fails to come to mind at the moment. Any advice?

Also, I'm not sure what the question is referring to with the term "upper bound." How does that value compare to the probability of the union of the four events?
A cheap upper bound is just the sum of the individual probabilities.

For that matter, a REALLY cheap upper bound is 47. Since no probability can be more than 1.

How tight an upper bound do you want?

If you want a tight upper bound, you have to say more about what the Ai's are, because you need to be able to subtract off the intersections.

Last edited by Maschke; August 31st, 2017 at 09:36 PM.
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September 5th, 2017, 07:10 AM   #3
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Yeah, a simple sum of the probabilities was all they were looking for. It should have occurred to me that's what they wanted, but it didn't. I appreciate the help!
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September 5th, 2017, 07:41 AM   #4
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I would be looking for a sum and the words "mutually exclusive" in any answer. Perhaps also a Venn diagram illustrating that mutual exclusivity is an upper limit on the size of the probability space for the union of events.
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