# Probability for events that can not be happened together

In a probability experiment, A, B and C are independent events. The probability that A will occur is r, the probability that B will occur is s, and the probability that C will occur is t whereas the r, s , and t are grater than zero.
Now I want to find a probability that, either A, B or C can occur but they can not occur together (like AAB or AAC is not allowed but AAA or BBB is allowed.)

In this case I used exclusion-inclusion formula to calculate the probability in this way:

$$P = P(A \cup B \cup C) - P(A\cap C) - P(B\cap C) -P(A\cap B) -P(A\cap B\cap C)$$

Is it a correct way?

• It almost sounds as if you are asking for the probability that exactly one of A, B, C occurs, but then the AAB not allowed but AAA allowed makes me wonder what the problem is about, – André Nicolas Dec 12 '15 at 6:44
• Yes only one of A, B or C will occur. – Rayan Ahmed Dec 12 '15 at 6:51
• I don't think the formula in the OP is right, but it is late and my blood caffeine level is low. The probability of exactly one is $r(1-s)(1-t)+s(1-r)(1-t)+t(1-r)(1-s)$. – André Nicolas Dec 12 '15 at 6:58
• @ André Nicolas, you are right and I got you that how you have done it but how to do it using set formula. – Rayan Ahmed Dec 12 '15 at 7:10

$P = P(A \cup B \cup C) - P(A\cap C) - P(B\cap C) -P(A\cap B) +2*P(A\cap B\cap C)$
• There will 2 in front of $P(A\cap B\cap C)$ – Rayan Ahmed Dec 12 '15 at 7:54