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Q: Santa offers you to play a game of dice. You get to roll a dice six times. You can stop rolling whenever you wish and you get the dollar amount shown on that roll. What is an optimal strategy to maximize your payoff?

Probability and Statistics (4th Edition)

A: Let us take a moment and think through this. At each point in the sequence of rolls you make, you have a decision to make. Do you keep rolling or do you stop and walk away with what is being "offered" to you? You also need to bear in mind that if you keep pushing your luck you will reach a point (the 6th roll) where you would have to be content with whatever comes out for the last roll. So lets start with the simple case of what the expected payoff is for the last roll. Lets call this \(E_{6}\). To compute it, simply take the payoff multiplied by the respective probability.

$$

E_{6} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{7}{2} = 3.5

$$

The general strategy to be followed is to check what the…

Q: Santa offers you to play a game of dice. You get to roll a dice six times. You can stop rolling whenever you wish and you get the dollar amount shown on that roll. What is an optimal strategy to maximize your payoff?

Probability and Statistics (4th Edition)

A: Let us take a moment and think through this. At each point in the sequence of rolls you make, you have a decision to make. Do you keep rolling or do you stop and walk away with what is being "offered" to you? You also need to bear in mind that if you keep pushing your luck you will reach a point (the 6th roll) where you would have to be content with whatever comes out for the last roll. So lets start with the simple case of what the expected payoff is for the last roll. Lets call this \(E_{6}\). To compute it, simply take the payoff multiplied by the respective probability.

$$

E_{6} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{7}{2} = 3.5

$$

The general strategy to be followed is to check what the…