Question

I know that if you have two probabilities \(P_A\) for event A and \(P_B\) for event B, the probability of event A or event B happening is \(P_A+P_B-P_AP_B\). What would the probability be for the probability of event A or event B or event C? Would it be by following the pattern, as\[P_A+P_B+P_C-P_AP_BP_C,\] or by iterating the previous one, as in the following?\[\left( P_A+P_B-P_AP_B\right) + P_C - \left( P_A+P_B-P_AP_B\right)P_C\]

Answer 1

It would be the latter. You have to subtract out all cases where multiples happen. Expanded it is \[
P_A+P_B+P_C-P_AP_B-P_AP_C-P_BP_C-P_AP_BP_C
\]

Answer 2

it is more complicated than that

Answer 3

Think about it this way, you have to add together all the probabilities, then subtract the ones that you are double-counting

Answer 4

\[P_A+P_B+P_C-P_AP_B-P_AP_C-P_BP_C+P_AP_BP_C\]

Answer 5

Oh right, got a sign flipped, whoops.

Answer 6

you have to add the intersection of all three because you have subtracted it off 3 times, so you have to put it back

Answer 7

if you draw a venn diagram you will see it. you have to arrange it so each portion is shaded exactly once

Answer 8

Interesting...wouldn't have expected that adding of the \(P_AP_BP_C\) term. Thanks.