Determining Optimal Strategy for Rolling Dice

In summary: If it was not from these, then you should not re-roll and take the first roll as your final value.In summary, if you are given the opportunity to re-roll after your first throw and the expected winnings from a re-roll are greater than the expected winnings from the original throw, then you should re-roll.
  • #1
bob j
22
0
Hi All,
I was wondering if anyone could give me a suggestion to solve this

"With one die, suppose in a round, you earn the amount of dollars equal to the value of the upwards face of the die. (eg. you earn $6 if you roll a six.) Now also suppose after your first roll, you are given the opportunity to cancel your first and roll again, taking that value as the final value. What should your strategy be? What if you are allowed to roll the die three times?"

thank you,
 
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  • #2
What are you expected winnings on a single roll of the die? Clearly this is $3.50, since all results 1-6 are equally probable.

If you get to roll a second time, you should take it if and only if your expected winnings are higher by rolling again -- that is, if you roll less than 3.5. This happens precisely when you roll a 1, 2, or 3, so your strategy is to roll again in precisely these cases. This makes your average winnings $(3.5 + 3.5 + 3.5 + 4 + 5 + 6) / 6 = $4.25.

If you get to roll three times, you should only take the first reroll if you could get better than 4.25.
 
  • #3
bob j said:
Hi All,
I was wondering if anyone could give me a suggestion to solve this

"With one die, suppose in a round, you earn the amount of dollars equal to the value of the upwards face of the die. (eg. you earn $6 if you roll a six.) Now also suppose after your first roll, you are given the opportunity to cancel your first and roll again, taking that value as the final value. What should your strategy be? What if you are allowed to roll the die three times?"

thank you,

If the first throw is a number from {1,2,3}, then your expected winnings if you roll again is $3.5 so you should cancel and roll again.

If you are allowed to throw 3 times and take the maximum from those 3 (let it be Y), you should observe that Y follows the following discrete distribution:

P(Y=1)=1/216
P(Y=2)=7/216
P(Y=3)=19/216
P(Y=4)=37/216
P(Y=5)=61/216
P(Y=6)=91/216

E(Y)= 4.96

Your strategy should be to re-roll if your first roll was from {1,2,3,4}.
 

1. What is the importance of determining the optimal strategy for rolling dice?

Determining the optimal strategy for rolling dice is important because it can help increase your chances of winning in a game that involves rolling dice. It can also help you make more informed decisions and minimize any potential losses.

2. How do you determine the optimal strategy for rolling dice?

The optimal strategy for rolling dice can be determined by calculating the probabilities of different outcomes based on the number of dice being rolled, the number of sides on each die, and any additional rules or limitations of the game. This information can then be used to make strategic decisions.

3. Can the optimal strategy for rolling dice guarantee a win?

No, the optimal strategy for rolling dice cannot guarantee a win. It can only increase your chances of winning by making more informed decisions based on the probabilities calculated. There is always an element of chance involved in rolling dice.

4. How can the optimal strategy for rolling dice be applied in real-life situations?

The optimal strategy for rolling dice can be applied in various real-life situations, such as board games, gambling, and even decision-making processes in business or daily life. By understanding the probabilities and making strategic decisions, one can increase their chances of success.

5. Are there any limitations to determining the optimal strategy for rolling dice?

Yes, there are limitations to determining the optimal strategy for rolling dice. These may include factors such as human error, changing game conditions, and the unpredictability of chance. It is important to remember that the optimal strategy is not a guaranteed win, but rather a way to improve your chances of success.

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