A business group is planning to build a Football Stadium in Halifax and wants to decide on the stadium capacity. Three capacity options have been suggested: small (S), medium (M) and large (L). Future payoffs depend on 1) the capacity of the stadium built, and 2) whether the local teams bid to join the Canadian Football League (CFL) is successful or not.

If the team bid is successful, the payoffs for a small, medium and large stadium are $50, $150 and $200 million, respectively. However, if the bid fails, these payoffs will become $50, $20 and $0 million, respectively. The group initially estimates a success probability of 20%.

To make a better guess about the success probability, the business group asked a specialist to give an expert's opinion. The specialist's track record of predicting similar outcomes indicates that he correctly predicted a success 80% of the time, but correctly predicted a failure only 60% of the time. In other words, given the outcome is a success, the specialist gave a positive opinion (PO) 80% of the time, and given the outcome is a failure, the specialist gave a negative opinion (NO) 60% of the time.

1. Which capacity option should be selected according to Bayes' decision rule (based only on the prior probabilities, before asking the specialist)?

2. Compute the posterior probabilities of success and failure conditional on the expert's opinion (PO or NO).

3. Use these probabilities to compute the expected payoff for each of the capacity options conditional on the expert's opinion, and determine the best capacity for each case.