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Joe has three coins C1, C2, and C3, with p1, p2, and p3 as their corresponding probabilities of landing heads. Joe flips coin C1 twice and then decides, based on the outcome, whether to flip coin C2 or C3 next. In particular, if the two C1 flips come out the same, Joe flips coin C2 three times next. However, if the C1 flips come out different, she flips coin C3 three times next. Given the outcome of Joe's last three flips, we want to know whether her first two flips came out the same.

What is the solution to this problem, assuming that p1 = 0.4, p2 = 0.6, and p3 = 0.1, and the last three flips came out as follows:

(ii)tails, heads, tails

(iii)tails, tails, tails