There are two problems in this HW, one using a decision tree and the other using the weighted average rule: wf + (1-w) q. Remember that q is your initial probability. N measures your confidence in q. If observing m non-events would make you change q to q/2, then N=m. If you have n observations, then f is the fraction of those observations in which the event of interest occurred and w=n/(n+N).

Weighted Average Problem:

A pharmaceutical company is considering running clinical trials on a drug designed to cure diabetes.

(1) Your experts estimates that there is an 80% probability of the drug being successful. But you recognize that your expert might be wrong. In fact, you would be willing to change your expert's estimate 80% probability to 40% if the drug were tested in four patients and, in every case, had failed to cure diabetes. While this example of four failures is purely hypothetical, it does provide a way to quantify your willingness to change the expert estimate of 80% based on new data.

What is your subjective probability for the drug being successful and what is your confidence, N, in that subjective probability?

(2) You won't produce the drug until there is a 95% probability of the drug curing a typical patient. Consider this second hypothetical case: Suppose the drug was given to k diabetics and all of them were cured. What is the smallest value of k which would lead you to revise your probability, using the weighted average rule, to equal or exceed 95%. (10 points)

(3) Suppose the drug was actually tested on six diabetics and five of them were cured. Using the previous value of N, the subjective probability and the weighted average rule, what is your estimate of the probability of the drug, if given to a diabetic, curing that patient? (5 points)