Explain the math behind each answer

4) Adam is a student in MS&E 252 who has just become the owner of the following deal: he can choose whether a thumbtack or a dollar coin will be tossed by the Professor. If he chooses a coin, he can then call whether it will land on its head or on its tail; if he chooses the thumbtack, he can call whether it will land with the point up or point down. If he correctly calls the toss of the coin or thumbtack he will win $100, otherwise he wins nothing. He is risk averse for these dollar prospects. He believes that the coin will land heads with probability 0.5. He is uncertain about the probability X of the thumbtack landing with the point up, with E[X] = 0.5 and Var[X] > 0. Consider the following statements: I. If Adam learns from the teaching team that the past 3 years students have chosen coins and they have always landed heads, he should prefer the coin over the thumbtack and call it heads. I]. If Adam has access to a thumbtack identical to the Professor's thumbtack, he can experiment to learn about X and he should choose the thumbtack over the coin. Even if Adam doesn't experiment on a thumbtack, he should choose the thumbtack over the coin. IV. If the deal is changed to two tosses instead of one and calling whether both tosses end the \"Same" or \"Different\