I want to generalize a formula and I need your help with this. This is not my homework or assignment but I need to come up with a concise formula that fits my documentation.
Background for my problem:
Considering all events to be independent of each other,
Let the probability of $Event$ $0$ be $P_{0}$ and $Event$ $1$ be $P_{1}$ and so on …
Then the probability that two events occur simultaneously is : $P_0P_1$ . This is nothing but the area of intersection of two circles $P_0$ and $P_1$.
Continuing the same way,
The total probability of simultaneous occurrence in case of three events is:
$P_0 P_1 + P_0P_2 + P_1P_2 – 2 P_{0}P_{1}P_{2}$.
Also can be visualized by drawing three intersecting circles.
One clarification here: This gives me the total probability of any two events plus all the three occurring at the same time right?
I cannot visualize the formula by drawing circles anymore. How can the above formula be generalized to get the probability of simultaneous occurrences when there are 4, 5, … independent events.
I have seen that the inclusion-exclusion principle is the answer. But I am not able to get an intuition for it. The inclusion-exclusion principle gives the probability of 2,3,4 sets intersecting but isn't my question different?
I get this doubt because, probability of four independent events occurring simultaneous is: $P_0P_1P_2P_3$. But what I need is a general formula for the total probability of two or more independent events at the same time.
Can anyone of you please throw some light?
Yes indeed I meant that the probability of "two or more events". The answer you have given is very precise and the one I was looking for. Yes it is boring to try to visualize circles when the number exceeds three. Instead, in general if use 1 – ($w_0 + w_1$) then we should land up correctly given that the events are independent. Thank you so much.
Best Answer
$$1-\left(1+\sum_{i=1}^n\frac{P_i}{1-P_i}\right)\cdot\prod_{k=1}^n(1-P_k)$$