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.

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(a) Using Bayes’ decision rule, what is the optimal prediction, and what is the corresponding expected payoff?

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(b) Suppose now that you may observe a practice toss of the chosen coin before predicting. Use the corresponding Excel template to find the posterior probabilities for which coin is being tossed.

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(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?