Probability cheat sheet webpage

Union bound

Expected number of times I get a certain outcome for a set of random variables with the same sample space, but potentially different and dependent probability distributions

Imagine I have two random variables ($X$ and $Y$) each of which can have value $A$ or $B$. Imagine I want to know the expected number of As I get. This will be:

$E[\text{number of }A\text{s}]=1\cdot p(X=A\text{ and }Y=B)+2\cdot p(X=A\text{ and }Y=A)$ $+1\cdot p(X=B\text{ and }Y=A)$

$=(p(X=A\text{ and }Y=B)+p(X=A\text{ and }Y=A))$ $+(p(X=B\text{ and }Y=A)+p(X=A\text{ and }Y=A))$

$=p(X=A)+p(Y=A)$

And this result works whether $X$ and $Y$ are independent random variables or not. The only thing we require is that getting $X=A$ and $X=B$ are mutually exclusive (and similarly for $Y$).

Inclusion-exclusion principle

https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle

Boole's inequality