please write by keyboard not hand writtin with word format

Choose 3 different problem types that we see in the lesson and create a sample question for each of them. Then, solve the questions in detail as if you are explaining them to someone who does not know anything about the subject. At least 2 of the 3 sample questions must be about Markov Chain.

It is ideal if the answers are between 4 and 8 pages in total.

Sample Question (Exponential Distribution)

The time between supernova explosions in the known universe is exponentially distributed with an average of 1 month. According to Mr. Spock, the spaceship Atlantic's scientist, a supernova explosion may deform the structure of the Higgs Boson, and may cause the extinction of the entire universe, with a probability of 1 in 120 billion. Assuming that the age of the universe is about 18 billion years, calculate the probability that such an event will cause the universe to disappear in the next 12 billion years.

Sample Question (Transition Probabilities)

Any year on a planet in the Sirius star system is either economic growth or recession (constriction). If there is growth for one year, there is 80% probability of growth in the next year, 20% probability recession is happening. If there is a recession one year, there is a 40% probability of growth and a 60% probability of recession the next year.

(a) If recession is known in 2265, find the probability of growth in 2267.

(b) What is the probability of a recession on the planet in the year Captain Kirk and his crew first visited the planet ?