1.Analyze the "Monty Hall Problem" which is stated as follows:There are three doors, numbered 1, 2, and 3. Behind one, chosen at random, is a new Ferrari and behind the other two are goats.You initially pick door 1.The game show host, Monty Hall, who knows where the Ferrari is, then opens one the remaining doors that has a goat behind it (there must be at least one), and shows you that this door does not contain the Ferrari. If the Ferrari is actually behind door 1 he opens either 2 or 3 with equal probability while if the Ferrari is not behind door 1 he has only one option.He then gives you the choice to either stay with door 1 or switch to the other unopened door.We want to decide what the best strategy is:stay or switch?

a.Make a tree diagram for this situation where the first level says what door the Ferrari is actually behind and then the second level indicates which door Monte opens.Label the diagram as usual.

b.If you adopt the strategy of staying with door 1 find your probability of winning.

c.If you adopt the strategy of switching to the other unopened door, what is your probability of winning?

2.Suppose that Martin, a student, knows 80% of the material to be covered on an upcoming True-False test.We take this to mean that on any randomly chosen test question there is a .8 probability that he will know the answer and we assume that in such a case he will give the correct answer.Also, we assume that if he does not know the answer to a particular question, he will guess and will have a .5 probability of getting it right.Suppose that a question is chosen at random.

a.What is the probability that Martin will give the correct answer to the question?

b.What is the probability that Martin knew the answer to the question, given that he gave the correct answer?