Final answer:
To find probabilities involving independent events A, B, and C, use the complement rule, the subtraction rule, and conditional probability. These rules help determine the probabilities of at least one, at most two, only one, and only A occurring. Example formulas are provided to calculate these probabilities.
Step-by-step explanation:
a) To find the probability that at least one of A, B, or C occurs, we can use the complement rule. The complement of at least one of A, B, or C occurring is none of them occurring. So, P(None of A, B, or C) = P(A' and B' and C') = P(A')P(B')P(C') = (1 - P(A))(1 - P(B))(1 - P(C)). Therefore, the probability that at least one of A, B, or C occurs is 1 - P(None of A, B, or C).
b) To find the probability that at most two of A, B, or C occur, we can use the complement rule again. The complement of at most two of A, B, or C occurring is three of them occurring. So, P(Three of A, B, or C) = P(A and B and C) = P(A)P(B)P(C). Therefore, the probability that at most two of A, B, or C occur is 1 - P(Three of A, B, or C).
c) To find the probability that only one of A, B, or C occurs, we need to find the probability of each event occurring individually and then subtract the probabilities of the other two events occurring. So, P(Only A occurs) = P(A) - P(B and A) - P(C and A). Similarly, P(Only B occurs) = P(B) - P(A and B) - P(C and B), and P(Only C occurs) = P(C) - P(A and C) - P(B and C).
d) To find the probability that only A occurs, we need to find the probability of A occurring and the probabilities of B and C not occurring. So, P(Only A occurs) = P(A)P(B')P(C').
e) To find the probability that A occurs given that only one of A, B, or C occurs, we need to find the conditional probability of A given that only one event occurs. So, P(A|Only one of A, B, or C occurs) = P(A and Only one of A, B, or C occurs) / P(Only one of A, B, or C occurs).