The principle of redundancy is used when system reliability is improved through redundant to backup components. Assume that your alarm clock has a 0.9 probability  of working on any given morning.  If you have two such alarm clocks, what is the probability that they both fail on the morning of an important final exam a)0.9 b)0.81 c)0.01 d)0.1

Hi! I pretty much don't understand this problem that was presented on my Statistics class

Based on a study from the Chronicles of Flippin'' Awesomeness, the probability that Napoleon and Pedro make it to their first period class on time is 0.29.  The probability that Napoleon and Pedro catch the bus is 0.87. However, the probability that they make it to their first period class on time, given that they catch the bus is 0.81.  What is the probability that Napoleon and Pedro catch the bus and make it to their first period class on time?

Please provide steps for how you got the solution to the problem provided below. A machine is used to produce one of each of two types of product (A and B) on alternating days, i.e., if A is produced today then B will be produced tomorrow and vice versa. Each day, there is a probability that the machine malfunctions. Malfunction probability after a day of operation is 0.03 when producing A and 0.07 when producing B. Once the machine malfunctions, it is no longer operable and will not be replaced until sometime in the future (beyond modeling horizon). Model the system as a Markov chain (give transition matrix on your paper). Suppose that on Monday product A was produced. Find the probability that the machine is still operable by the morning of Thursday. Your model should have a state ”malfunction”, which cannot transition to any other state other than itself.

Based on a study from the Chronicles of Flippin'' Awesomeness, the probability that Napoleon and Pedro make it to their first period class on time is 0.30.  The probability that they make it to their first period class on time, given that they catch the bus is 0.90.  The probability that Napoleon and Pedro catch the bus and make it to their first period class on time is 0.21.  What is the probability that Napoleon and Pedro catch the bus?