I am trying to understand conditional probability. In this numerical, L is the event of being late to office, T is the event of heavy traffic and R is the event of raining.
I can solve this problem based on the formula. But I am trying to comprehend what does L∩T∩R and (L|T∩R) mean.
This is the main source of the numerical I am looking into - Link
(I removed compliment to avoid confusion)
For events $A$ and $B$, $P(A|B)$ is defined by the following relationship: $$P(A|B) = \frac{P(A\cap B)}{P(B)}$$
Thus,$$P(L|T\cap R)=\frac{P(L\cap T\cap R)}{P(T\cap R)}$$
So the relationship between the two expressions is that factor of $P(T\cap R)$. But since you are strictly talking about sets/events and not the probabilities of them, a more direct answer is that $L\cap T\cap R$ is the event of being late to the office and there being heavy traffic and rain, while $L|T\cap R$ is the event that given that there's heavy traffic and it's raining, that you are late to the office.