2. A student's alarm clock fails to go off 1 time out of 3. In addition, the student misses the bus with probability 1/2 if his alarm clock fails. If his alarm clock works, he misses the bus with probability 1/3. If the student's alarm clock fails and he misses the bus, he arrives late to class with probability 1/2. This probability is reduced to 1/6 if the alarm clock works and he does not miss the bus. In all other situations (i.e., when the alarm clock fails and he does not miss the bus, and alarm clock works and he misses the bus), the probability of the student being late to class is 1/4. (a) On a random day, what is the probability that the student's alarm clock works, he does not miss the bus and he is not late to class? Make sure to clearly define all notation used. (b) What is the probability that this student arrives late to class? (c) What is the probability that the alarm clock failed, given that he arrived late to class