Suppose there is a particular stock that has only a $10\%$ chance of going up$.$ Someone is considering investing big, investing small, and not investing at all. If they invest big, the payoff will be $$200$ if the stock goes up and the $$$$50$ if it goes down. If they invest small, the payoff will be $$30$ if the stock goes up and the $$$$7\mathrm{.}5$ if it goes down. Obviously, not investing has a certain payoff of $$0.$

a$)$ Construct the decision tree and calculate its expected value.

b$)$ What is the value of perfect information for this problem?

c$)$ What is the value of perfect control for this problem?

d$)$ Suppose, a particular financial advisor is able to correctly predict what a stock will do $90\%$ of the time. We are considering hiring this analyst to make a prediction for us about this stock. Find the value of the imperfect information that the analyst can provide.

e$)$ Suppose the financial advisor offers one free prediction for a new customer, and does so here. The advisor predicts the stock will go up$.$ Now what is the value of perfect information?