The probability of an event is given by the number of favorable outcomes divided by the total number of outcomes. Here the event is the alarm clock running out of power. There are 24 hours during which the power can go out. You are asleep during 8 of these. A "favorable" outcome in this case is the power going out while you are sleeping -- that is, during one of those 8 hours. This makes the probability that the power goes out while you are sleeping 8/24.
