Those damn dice... (Part II)

Last time  we investigated questions that ask about the probability of a variety of dice rolling situations. This time we focus on games that involve dice.

 DR04
You roll a six-sided die and a ten-sided die. If you guess the sum correctly you win an amount that equals the sum. What is the optimal guess?

DR04 - Answer:
First of all, think about what "optimal guess" means. It should maximize the expected payoff. Hence we can iterate a table:
Sum                   | 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
# combination  | 1  2  3  4  5  6  6  6   6   6   5   4   3   2   1
The product is maximized at sum = 11, which is the answer.

DR05
What is the expectation value of the product of rolling a six-sided die and a eight-sided die?

DR05- Answer:
Since the two rolls are independent events, E[A & B] = E[A] * E[B] = 15.75

DR06
The game is as follow: Roll a 12-sided die. You can either get the amount of the roll, or choose to roll two 6-sided dice, at which point you must receive the sum of the two dice. What is the game worth?

DR06- Answer:
Use backward induction. The expected payoff of the second round is 7. Hence you would go for round 2 only if the first round was less than or equals 6. Thus the payoff is
(7 + 8 + 9 + 10 + 11 + 12) / 12 + 0.5 * 7 = 8.25


DR06
What is the expectation value of the absolute difference of the points on two 6-sided dice?

DR06- Answer:
By simple state space counting, the answer is
(0 * 6 + 1 * 10 + 2 * 8 + 3 * 6 + 4 * 4 + 5 * 2) / 36 = 1.94

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