Please show all steps to derive the new generalized addition

Please show all steps to derive the new generalized addition rule for the given 3 events A,B,C so that I may understand each step.

What will the A,B,C version be?

Thanks

Solution

The generalized addition rule would be

P(A U B U C) = P(A) +P(B) + P(C) - P(A and B) - P(B and C) - P(A and C) + P(A and B and C) [ANSWER]

where \"and\" = the inverted U, the intersection.

We actually derive this by inspecting a Venn Diagram, and counting its parts.

We add the three events, then subtract those which got double counted, then compensate for what was lost by subtracting.