Suppose a new type of artificial heart has been developed. The heart has three moving parts that are subject to failure. In a given time interval the probability that component A fails is 0.1, the probability that component B fails is 0.15, and the probability that C fails is 0.05. Assume that these failure events are independent.
a) What is the probability that at least one component fails in the time interval?
b) Given that at least one component failed, what is the probability that component A failed?
c) If both components A and B fail, the heart will fail. If component C fails, the heart will fail. What is the probability that A failed given that the heart has failed?