Probability of a Union Proof

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Brooklyn
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TL;DR Summary
Probability of a Union using Indicator Functions
"Prove Theorem 7.1 about the probability of a union, using the 12.3 proof (see section 12.2) that involves indicator variables. Do not write the proof in full generality, only for three events. You should not use the product notation; you should write out all factors of the product."

I'm taking a calculus-based intro to probability and stats course that's not intended for math majors. I have a professor who is terrible at teaching and expects that students should easily be able to do the proof. I asked for help and he told me that it'd make sense if I worked out an example. I'm not sure how to work out an example if I don't understand the proof. None of the students in the class understand the proof.

During class, he reads from his notes (excerpts below) and doesn't work out examples. A month into the course, he says we need more theory before he supposedly gets to examples. I found nothing on the net to explain the proof. Any help would be greatly appreciated and I'd pass it on to the rest of the class which is also lost.

Screen Shot 2020-09-24 at 9.14.56 PM.png

Screen Shot 2020-09-24 at 9.13.53 PM.png
 

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  • #2
StoneTemplePython
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they're doing inclusion-exclusion... what exactly is your question? I think this is worked out in a more friendly way in the free book by Blitzstein and Hwang https://projects.iq.harvard.edu/stat110/home

For the second approach with indicators: it helps to know what an elementary symmetric function is and how to factor or expand a polynomial
 
  • #3
Brooklyn
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I'm trying to figure out how to write my own version of the 12.3 proof for "two, three, four, or five events." I tried to ask if someone could help write a proof for 3 events, then I could work out the other cases. I think the notes provide the general proof and we're supposed to translate that.

Thanks for the link, I'll lookup inclusion-exclusion in the book.
 
  • #4
StoneTemplePython
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why don't you do the proof for ##n=2## items? Draw a venn diagram and pay attention to what you are double counting...

Once you've mastered ##n=2##, try ##n=3## which is very doable. ##n=4## may be workable but it starts to get a bit tedious around ##n\geq 4## and some abstraction is needed.
 
  • #5
Stephen Tashi
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"Prove Theorem 7.1 about the probability of a union, using the 12.3 proof (see section 12.2) that involves indicator variables. Do not write the proof in full generality, only for three events. You should not use the product notation; you should write out all factors of the product."

I interpret that to mean that your write-out the proof of Theorem 7.1 for the special case ##n = 3##.

For example, instead of ##\Pi _{i=1}^{n} (1 + (-1)I_i)##, you write ##\Pi_{i=1}^{3} (1 + (-1)I_i)) = (1 - I_1)(1-I_2)(1-I_3) = ## whatever eq. 3.3 says in this case.
 

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