Candidates Smith and Johnson are running for US President. Peter the Pundit estimates the probability that Johnson wins in each of the 51 states (let us call Washington, DC also a state). For the sake of example, assume that each state sends only one person to Electoral College. The person who gets a simple majority wins. There are 30 strongly pro-Smith states: Johnson wins in each of these states only with probability 20%. The other 21 states are strongly pro-Johnson: Johnson wins in each of them with probability 90%. All states vote for Smith or Johnson independently of each other. A political weekly asks Peter: "Who will win in each state?" For pro-Smith states, Peter says "Smith". For pro-Johnson states, Peter says "Johnson".
(i) Approximate using Poisson distribution the number of states that Peter is going to get wrong. What is the probability that he gets all states right?
(ii) Estimate the probability that Johnson wins, using Normal approximation.