Problem 1) In a communication system with binary input X and binary output Y, the transmitted signalX0,1with probability P(X=0)and P(X=1) = 1-p.The transmission channel can be modeled by the following conditional probability: P(Y=0 | X = 1) = qand P(Y=1|X=0) = r.Find the probability P(Y=0)and simplify the result if q=r .

Problem 2)The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year.

a. What is the probability that there will be exactly 3 power outages in a year?

b. What is the probability that there will be at least 3 power outages in a year?

c. What is the probability that the number of outages is between 1 and 3 (1 and 3 inclusive)?

Please include the steps involved with formulas and how the decision was arrived in clear steps about do it.