# How do you solve for the intersection or multiplication rule of three dependent events?

Mathematics Asked by AndroidV11 on September 27, 2020

Yes I am not aware of how to edit in LaTeX or make elegant edits, so apologies for not being able to do such editing on my own.

I am aware that $$P(A cap B)$$ for dependent events has $$2$$ variants based on the the $$2!$$ number of permutations in which $$A$$ and $$B$$ can be written.

$$P(A cap B) = P(A | B) P(B)$$

$$P(B cap A) = P(B | A) P(A)$$

My questions are,

1. What is the equivalent of this for three events given $$A, B$$ and $$C$$?
2. If I am right to assume, can we make $$3!$$ or $$6$$ versions of such a formula? What would they look like?
3. This one does not need me to see the formula of permuted variants, but what would the intersection of $$n$$ sets look like in some summary notation like summation?

I am particularly interested in wanting to see a sample problem and solution involving the intersection of three dependent events, but I don’t think this site would find it nice if someone just goes around asking for x type of problem and solution even if it was simple just for me to get an illustration. But if it is allowed within the rules, I would appreciate it.

If $$mathbb P(Acap B)ne 0$$ then $$mathbb P(Acap Bcap C)=mathbb P(A) mathbb P(Bmid A) mathbb P(Cmid Acap B)$$ And there are $$3!=6$$ such versions. For $$n$$ events $$A_1,ldots, A_n$$, if $$mathbb P(A_1capldotscap A_{n-1})neq 0$$, $$mathbb P(A_1capldotscap A_n)=prod_{i=1}^n mathbb Pleft(A_iBiggm | bigcap_{j=0}^{i-1} A_{j}right)$$ where it is assumed that $$A_0=Omega$$.