Consider the Casino Problem presented in the lectures, but with six plays of the game, and the goal being to accumulate at least six chips, beginning with 3 chips, where the probability of winning at each play of the game is 55%. In the Dynamic Progarming model with results presented below, the recursion is forward, i.e., the stages range from n=1 (first play of the game) to n=6 (final play of the game). The state is the number of chips accumulated, and the decision is the number of chips to bet at the current play of the game.

a) Compute the missing number in the table for stage 1.

b) What is the probability that six chips can be accumulated at the end of six plays of the game? (%)

c) How many chips should be bet at the first play of the game? (If more than one value is optimal, choose an answer arbitrarily.)

d) If one bets the amount you selected in (c) and the first play of the game is won, what should be the bet at the second play of the game? If the first play of the game is lost, what should be the bet at the second play of the game?

113456 Max I o.00 I 0.00 I 0.55 4 0.00 0.00 0.55 0.55 0.55 xxxxxxxxx 0.55 5 0.00 0.55 0.55 0.55 0.55 0.55 XXxX 0.55 6 1.00 0.55 0.55 0.55 0.55 0.55 0.551 1.00 10.00 0.00 210.00 0.00 0.00 3 0.00 0.00 0.00 0.55 10.00 0.00 2 I 0.00 0.30 0.30 xxxxxxxxxxxxxxxxx1 o.30 3 0.55 0.30 0.30 0.55 xxxxxxxxxxxxxxI 0.55 4 0.55 0.55 0.55 0-55 0.55 XXXXXxxxxI 0.55 5 0.55 0.80 0.80 0.55 0.55 0.55 xxxxI 0.80 6 I 1.00 0.80 0.80 0.80 0.55 0.55 0.551 1.00 I 0.00 s1x:12345I MAX 00.00 xxxxxxxxxxXXKXExKXxxxxxx 0.00 l 0.17 I 0.30 I 0.55 4 I 0.55 0.69 0.69 0.55 0.55 xxxxxxxxx .6S 5 10.90 0.90 0.80 0.69 0.55 0.55 xxxxI 0.80 6 I 1.00 0.51 0.80 0.80 0.69 0.55 0.551 1.00 1 10.00 0.17 2I0.30 0.30 0.30 3 I 0.55 0.44 0.44 0.55 -Stage 3 10.00 1 0.17 0.17 2 10.30 0.38 0.38 3 0.55 0.51 0.51 0.55 4 1 0.69 0.69 0.69 0.62 0.55 xxxxxxKI 0.69 5 1 0.80 0.86 0.80 0.69 0.62 0.55 xxxxI 0.6 0.17 xxxxxxxxxxxxxxxxxxX 0.38 1 0.55 110.17 0.21 XXXXXXXXXXXXXXXXXXXXXXXXI 0.21 3 0.55 0.55 0.55 .55 XXXXXXXXXXXXX 0.55 4 10.9 0.72 0.72 0.62 0.55 xxxxxxxxXI 0.72 5 10.86 0.86 0.80 0.72 0.62 0.55 XXxXI 0.86 61.00 0.94 0.86 0.80 0.72 0.62 0.551 2.00 tage 1- 0.57 0.55 0.57 113456 Max I o.00 I 0.00 I 0.55 4 0.00 0.00 0.55 0.55 0.55 xxxxxxxxx 0.55 5 0.00 0.55 0.55 0.55 0.55 0.55 XXxX 0.55 6 1.00 0.55 0.55 0.55 0.55 0.55 0.551 1.00 10.00 0.00 210.00 0.00 0.00 3 0.00 0.00 0.00 0.55 10.00 0.00 2 I 0.00 0.30 0.30 xxxxxxxxxxxxxxxxx1 o.30 3 0.55 0.30 0.30 0.55 xxxxxxxxxxxxxxI 0.55 4 0.55 0.55 0.55 0-55 0.55 XXXXXxxxxI 0.55 5 0.55 0.80 0.80 0.55 0.55 0.55 xxxxI 0.80 6 I 1.00 0.80 0.80 0.80 0.55 0.55 0.551 1.00 I 0.00 s1x:12345I MAX 00.00 xxxxxxxxxxXXKXExKXxxxxxx 0.00 l 0.17 I 0.30 I 0.55 4 I 0.55 0.69 0.69 0.55 0.55 xxxxxxxxx .6S 5 10.90 0.90 0.80 0.69 0.55 0.55 xxxxI 0.80 6 I 1.00 0.51 0.80 0.80 0.69 0.55 0.551 1.00 1 10.00 0.17 2I0.30 0.30 0.30 3 I 0.55 0.44 0.44 0.55 -Stage 3 10.00 1 0.17 0.17 2 10.30 0.38 0.38 3 0.55 0.51 0.51 0.55 4 1 0.69 0.69 0.69 0.62 0.55 xxxxxxKI 0.69 5 1 0.80 0.86 0.80 0.69 0.62 0.55 xxxxI 0.6 0.17 xxxxxxxxxxxxxxxxxxX 0.38 1 0.55 110.17 0.21 XXXXXXXXXXXXXXXXXXXXXXXXI 0.21 3 0.55 0.55 0.55 .55 XXXXXXXXXXXXX 0.55 4 10.9 0.72 0.72 0.62 0.55 xxxxxxxxXI 0.72 5 10.86 0.86 0.80 0.72 0.62 0.55 XXxXI 0.86 61.00 0.94 0.86 0.80 0.72 0.62 0.551 2.00 tage 1- 0.57 0.55 0.57