1. Suppose that you are considering an idea (v,c). There is uncertainty over the true

value v. Looking at similar ideas from the past, you estimate that the probability that v = 1000 is

1/4 and the probability that v = 200 is 3/4. With certainty, you know that c = 500 and R = 1.

a. What is the expected value of v?

b. Should this idea be invested in given the above probabilities? Why or why not?

c. Now, suppose that you decide to consult an expert to better determine whether v =

1000 or v = 200. The expert excels at confirming high-value ideas but is rather poor at

confirming low-value ideas. Suppose that when the true value is v = 1000, the expert

always correctly declares the value as v = 1000. However, when the true value is v = 200,

the expert incorrectly declares the value at v = 1000 with probability 1/2 and correctly

declares the value at v = 200 with probability 1/2. Use Bayes' rule to determine the

probability that v = 1000 when the expert declared that the value is v = 1000.

d. Using your answer to part c, what is the corresponding expected value of v?

e. Should this idea be invested in given the updated probabilities? Why or why not?

f. Compare your answers to b and e. What does this suggest about the importance of

Bayes' rule and updating probabilities when new information is available?

2. Consider two ideas (v,c), where idea 1 is (5,25), and idea 2 is (4,15). Suppose that, at

most, one idea may be selected. We are using the efficiency of the ideas to determine which

idea should be invested in.

a. If R = 2, which idea should be selected?

b. If R = 4, which idea should be selected?

c. If R = 8, which idea should be selected?

d. If R = 16, which idea should be selected?

e. If R = 32, which idea should be selected?

f. What do these answers suggest regarding the importance of c relative to v as R

increases?

3. Consider three ideas (v,c), where idea 1 is (5,25), idea 2 is (4,15), and idea 3 is (6, 29).

Find a value of R such that:

a. Idea 1 is chosen.

b. Idea 2 is chosen.

c. Idea 3 is chosen.

If such a value R does not exist for any of the above, briefly explain why.