Question

A particulate monitor has a power supply consisting of two batteries in parallel. Either battery is adequate to operate the monitor. However, since the failure of one battery places an added strain on the other, the conditional probability that the second battery will fail, given the failure of the first, is greater than the probability that the first will fail. On the basis of testing it is known that 7% of the monitors in question will have at least one battery failed by the end of their design life, whereas in 1% of the monitors both batteries will fail during the design life.

(a) Calculate the battery failure probability under normal operating conditions.
(b) Calculate the conditional probability that the battery will fail, given that the other has failed.

Answer 1

The battery failure probability under normal operating conditions is 99%. The conditional probability that the battery will fail, given that the other has failed, is approximately 14.29%.

Step-by-step explanation:

To calculate the battery failure probability under normal operating conditions, we can use the principle of complementarity. The probability of at least one battery failing is equal to 1 minus the probability of both batteries operating properly. From the given information, we know that 7% of the monitors have at least one battery failed, and 1% have both batteries failed. Therefore, the probability of both batteries operating properly is 1% and the battery failure probability under normal operating conditions is 99%.To calculate the conditional probability that the battery will fail, given that the other has failed, we can use Bayes' theorem. Let A be the event that the second battery fails, and B be the event that the first battery fails. The conditional probability of A given B is equal to the probability of A and B occurring divided by the probability of B occurring. From the given information, the probability of A and B occurring is 1%, and the probability of B occurring is 7%. Therefore, the conditional probability that the battery will fail, given that the other has failed, is approximately 14.29%.