Parts (a), (b) and (d) are worth two points each. Part (c) is worth six points and part (e) is worth twelve points. Work together but hand in one per person.

Three people are playing paintball and are engaged in a three-way duel.The players are Juan, Emma, and Madison.There are two rounds. In the first round, Juan goes first. If Emma was not hit by Juan, then she gets to have a shot. If Madison is not hit by Juan or Emma, then Madison gets to have a shot.After the first round, anyone not hit by a paintball goes to the second round.

In the second round the order is as follows. If Juan and Emma are the only ones left, then Juan goes first.If Juan and Madison are the only ones left, then Juan goes first and if Emma and Madison are the only ones left then Emma goes first.

Juan is a poor shot, with only a 30 percent chance of hitting a person at whom he aims. Emma is a much better shot, achieving 80 percent accuracy. Madison is a perfect shot, never missing. Assume that each party engages in optimal behaviorwhich is to maximize the probability of surviving the two rounds.

1. If both Emma and Madison are in the game after Juan's first shot who will Emma aim at?
2. If both Emma and Madison are in the game after Emma's first shot who will Madison aim at?
3. In Round One what is Juan's optimal strategy? Explain why that strategy makes sense.
4. T/F.Explain. It is impossible for all three players to make it to the second round.
5. Assume that this game is played 100,000 times. Calculate the number of times that Juan has not been hit after two rounds, the number of times that Emma has not been hit after two rounds, and the number of times that Madison has not been hit after two rounds. Show all your work.