[Advanced] Probability of Union[n-elements]

[Bassalisk]

Hello,

We are all familiar with the formula that relates union of 2 mutually NOT exclusive events formula:

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

For 3 sets its easily derived using this formula.

But I wanted to take this step further. I wanted to find a general formula, that represents union of n elements.

I don't know how to write that In LaTex.

If anybody knows the answer, please don't tell me. Tell me some guidelines to solution. I have tried, but I get stuck with recursive sums, and I can't get out of them.

 

[Bacle]

Think of expressing a union of n elements as a union of a smaller number of elements,and then use the answer you already know, by using parentheses. I am trying to not be neither too obscure nor tell you the answer.

 

[Bassalisk]

Think of expressing a union of n elements as a union of a smaller number of elements,and then use the answer you already know, by using parentheses. I am trying to not be neither too obscure nor tell you the answer.

You mean like, doing for 3 4 5 and maybe 6 sets this union, then get my answer from that?

Is that mathematically bulletproof?

 

[micromass]

You mean like, doing for 3 4 5 and maybe 6 sets this union, then get my answer from that?

Is that mathematically bulletproof?

It's certainly not bulletproof, but it's a nice start. Start by finding it for 3,4,5 and 6 and see if you can generalize it. Once you've found a candidate for a general solution, then you can apply induction to prove it.

 

[Bassalisk]

It's certainly not bulletproof, but it's a nice start. Start by finding it for 3,4,5 and 6 and see if you can generalize it. Once you've found a candidate for a general solution, then you can apply induction to prove it.

Fun! On it

 

[Bassalisk]

Here is what I got so far:

$P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)$

Assuming that:

$P((A\cap B\cap C)\cup D)=P((A\cap D)\cup (B\cap D)\cup (C\cap D)$

then:

$P(A\cup B\cup C\cup D)=P(A)+P(B)+P(C)+P(D)-P(A\cap B)-P(A\cap C)-P(A\cap D)-P(B\cap C)-P(B\cap D)-P(C\cap D) +P(A\cap B\cap C)+P(A\cap C\cap D)+P(B\cap C\cap D)-P(A\cap B\cap C\cap D)$Ok, I see a sum here $\sum_{i=1}^{n} P(A_{i})$

I also see that each set is intersected with every other set. I don't know exactly how to write that.

At least not in LaTex. I am thinking:

$\bigcap_{i,j=1}^{n} A_{i},A_{j} i\neq j$ [idea]

I get stuck at those 3 intersections and 4.

 

[Bacle]

Now, try to see if you can detect a pattern and try induction. If you want, I can give you the (an) answer with a spoiler warning, for when you're done' let me know.