Proving Probability of Union with Indicator Variables in Three Events

[Brooklyn]

TL;DR

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.

 

[StoneTemplePython]

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

 

[Brooklyn]

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.

 

[StoneTemplePython]

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.

 

[Stephen Tashi]

"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.