1. We have a bag with a red and a blue ball, which we pick uniformly at random. If the red ball is chosen, we set W = 1, and throw a coin that has probability of 2/3 of being heads twice. If the blue ball is chosen, we set W = 0 and throw a coin that has probability of 1/3 of being heads twice. Let Fo = {0,2}, Fi be the o-algebra generated by the outcome of the first coin flip, and F2 be the o-algebra generated by the outcome of both coin flips. Calculate Wn = E[W|Fn] for n = {0,1,2}. Confirm explicitly (i.e. by checking that E[Wn+1|Fn] = Wn) that {W0,W1,W2} make up a martingale sequence with respect to the filtration En. 1. We have a bag with a red and a blue ball, which we pick uniformly at random. If the red ball is chosen, we set W = 1, and throw a coin that has probability of 2/3 of being heads twice. If the blue ball is chosen, we set W = 0 and throw a coin that has probability of 1/3 of being heads twice. Let Fo = {0,2}, Fi be the o-algebra generated by the outcome of the first coin flip, and F2 be the o-algebra generated by the outcome of both coin flips. Calculate Wn = E[W|Fn] for n = {0,1,2}. Confirm explicitly (i.e. by checking that E[Wn+1|Fn] = Wn) that {W0,W1,W2} make up a martingale sequence with respect to the filtration En