# What is the probability of multiple events occurring together?

I am analysing the probability of an employee leaving a company from multiple different reasons.

Say that the probability they leave because of a event a is p(a) and the probability they leave because of a event b is p(b) ... p(c) ... p(d)

How do we calculate the overall probability that they leave?

is we know that some of these events are independent and some are dependent, but not sure which ones, how would we go ahead finding the overall probability?

is it simply p(a) + p(b) + p(c) + p(d) = p(a) X p(b) X p(c) X p(d) ?

• The event that the employee leaves the office is either due to $a, ~b, ~c,$ or $d.$ So, ... Sep 28, 2022 at 12:08
• Can an employee leave for multiple reasons? If so, best to rephrase this as "$P(a)$ is the probability that $a$ was listed as a reason for leaving."
– Eoin
Sep 28, 2022 at 12:10
• If you want to know something like $P(a\text{ or }b\text{ or }c\text{ or } d)$ then you must seek your hail in PIE (i.e. the principle of inclusion/exclusion). Oct 5, 2022 at 14:11

Your problem seems to be ill-formulated. Your original formulation suggests that the employee left, and that you are computing the probability $$p(X)$$ that the event causing the employee to leave was $$X$$. The probability of event $$L$$ ("the employee leaves") is then 1.
I guess you are actually interested in the probability of $$L$$ conditioned on $$X$$. $$p(L|X)$$ is the probability that the employee leaves, knowing that $$X$$ took place, and $$p(X)$$ is the probability of $$X$$ taking place. Then $$p(L)$$ can be computed by the law of total probability:
$$p(L) = \sum_{X \in \mathcal{X}} p(L|X)p(X)$$