There are two biased coins with probabilities of landing heads of 0.8 and 0.4, respectively. One coin is chosen at random (each with probability 1/2) to be tossed twice. You are to receive $100 if you correctly predict how many heads will occur in two tosses.
(a) Using Bayes’ decision rule, what is the optimal prediction, and what is the corresponding expected payoff?
(b) Suppose now that you may observe a practice toss of the chosen coin before predicting. Use the Excel template for posterior probabilities to find the posterior probabilities for which coin is being tossed.
(c) Determine your optimal prediction after observing the practice toss. What is the resulting expected payoff?
(d) Find EVE for observing the practice toss. If you must pay $30 to observe the practice toss, what is your optimal policy?