probability question

Question

On the night before your final exam, you are nervous about your alarm not waking you up. You set several additional alarms as backup. Suppose you know that your phone alarm works 99% of the time, your alarm clock works 95% of the time, and your wrist watch alarm works 92% of the time. Assume all three alarms operate independently of each other. Setting all three, what is the probability that none of the alarms work (causing you to miss the exam)?

Answer 1

The probability that none of them work is the same as the probability that all of them don't work. So we need the probabilities of failure for each alarm.

P(phone fails) = 1 - 0.99 = 0.01

P(watch fails) = 1 - 0.95 = .05

P(alarm fails) = 1 - 0.92) = 0.08

Since the alarms are independent, then we multiply the probabilities.

(0.01)(0.05)(0.08) = 0.00004 = 0.004%