Answer:
21/12
Explanation:
Your expected payoff for a scenario that your first roll gives you v1 and second roll gives you v2 = P(v1)*P(v2)*v2. i.e., probability that your first roll returns v1 multiplied by probability that your second roll gives you v2, multiplied by final payoff (equal to v2). Also assuming that the dice is unbiased, probability of dice throwing up any value would be equal to 1/6
If you go with only a single roll, your expected payoff = P(v1) * v1
Now a good strategy will be to roll the dice first, see if your value is atleast 4. If you get 4 to 6, stop there and take the payoff because there is a 50% probability that your second roll will be between 1–3, which is definitely lesser payoff. Instead if your first roll is 1–3, go for second roll because there’s a 50% probability that your second roll gives 4–6, which is definitely higher payoff. With this strategy and the above formulation, lets see the payoffs in different scenarios:
Scenario 1: First roll gives 1 to 3:
P(first roll = 1 to 3) * P(second roll = 1) * 1 + P(first roll = 1 to 3) * P(second roll = 2) * 2 + ……. + P(first roll = 1 to 3) * P(second roll = 6) * 6 = (1/2) * (1/6 + 2/6 + 3/6 + …6/6) = 21/12
Scenario 2: First roll gives 4 to 6:
P(first roll = 4)* 4 + P(first roll = 5)* 5 + P(first roll = 6)* 6 = (1/6 * 4 + 1/6 * 5 + 1/6 * 6) = (4+5+6)/6 = 15/6
Total expected payoff = 21/12 + 15/6 = 51/12 = 4.25
A couple of sanity checks here would be:
The expected payoff would never be more than max. payoff, which is equal to 6. That’s because assume you devise a strategy where you get max payoff everytime (ex: you tamper the dice so it gives you 6 every time). Even in that case, your expected payoff would be 6. So for an unbiased dice, your expected payoff has to be less than or equal to 6
If you have to stop after one roll, the expected payoff = average of facevalue of the dice = (1+2+3+4+5+6)/6 = 3.5; You now have a leverage that you can either stop here, or go to the next roll. Any smart game will have a payoff of atleast this value (3.5)
So your answer will be something in between 3.5 and 6; Our current answer, 4.25 satisfies this.