The probability that the switch does not fail is 1-0.001=0.999. In order to switch the lights off and on 1200 times, we need to press the switch 2400 times. Each event is assumed independent from the others; when we press the switch, its probability that it will fail is not influenced by the previous pressing of the switch. Thus, the probability that it won't fail after 2 times is 0.999*0.999=0.999^2. Similarly, we get that the required probability is : [tex]P=0.999^{2400}=0.091[/tex]
This is a relatively low probability. We see that even if the probability that the switch fails one time is very low, after many repetitions it becomes quite probable that the switch will fail at least one time.
