Probability of Computers Crashing

Consider a system that has 4 computers. The system is down if at least one computer crashes. The system works if all 4 computers are working. During each day, the probability that a single computer crashes is 5%, independent of other computers and other days.

(a) (4 points) Let us first focus on one day. What is the probability that on this given day, the system works with no downtime, i.e. what is the probability that zero computers crash?

What is the probability that the system is down on a given day? (Hint: Think complements!)

Let us denote by p the probability of the link being down on a given day. In case you have doubts about computing p in part (a), you can solve parts (b)-(d) by using the symbol p instead of its actual value. Let Y be the random variable accounting for the number of days in a month (n=30 days) that the system is down.

(b) (3 points) What is the distribution of Y? What is its mean and standard deviation?

(c) (3 points) What is the probability that, in a given month, the system is down for at least one day, i.e., Pr(Y>=1)?
(Hint: You can obtain an exact answer by considering the complement!)

(d) (3 points) Give an approximation of the probability that, in a given month, the system is down for at least 10 days, i.e. P(Y>=10).
(Hint: Obtain an approximation via the Central Limit Theorem! Notice that random variable Y can be written as the sum of 30 random variables.)