$$P(A)\frac{2}{3}, P(A | B)= \frac{1}{3}$$ and $$P(A ∪ B)= \frac{4}{5}.$$ Find P(B).
I honestly have no idea how to even approach this problem, as I cannot find any helpful online notes on Combined probability. For instance, I don't even know how to find P(A ⋂ B) if given P(A ∪ B), P(A) and P(B).
So how would I solve a simple problem like finding P(A ⋂ B), and then how would I use this knowledge to solve the question mentioned above?
You don't need to find $P(A \cap B)$, but you can use the knowledge that $P(A \cap B) = P(A|B) \times P(B)$
By the Inclusion-Exclusion principle:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$\frac{4}{5} = \frac{2}{3} + P(B) - P(A \cap B)$
But $P(A \cap B) = P(A|B) \times P(B)$
So $\frac{4}{5} - \frac{2}{3} = P(B) - P(A|B) \times P(B)$
We know $P(A | B)$
$\frac{2}{15} = P(B) (1 - \frac{1}{3})$
$P(B) = \frac{1}{5}$