Probability: Dice Game between 20- and 12-sided dice with re-rolls

In summary: Instead, one should use a mathematical approach and calculate the probability of winning based on the choices of numbers to re-roll. This strategy will lead to an optimum solution for both players and a more accurate answer.
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Master1022
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Homework Statement
One person has a 12 sided die and the other has a 20 sided die. They each get two rolls and they can each chose to stop rolling on either one of the rolls, taking the number on that roll. Whoever has the higher number wins, with the tie going to the person with the 12 sided die. What is the probability that the person with the 20 sided die wins this game? Assume the players cannot see the others' roll.
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Hi,

I was reading around and found this problem. I have seen some discussion about the solution (but nothing verified) with some disagreement.

Problem: One person has a 12 sided die and the other has a 20 sided die. They each get two rolls and they can each chose to stop rolling on either one of the rolls, taking the number on that roll. Whoever has the higher number wins, with the tie going to the person with the 12 sided die. What is the probability that the person with the 20 sided die wins this game? Assume the players cannot see the others' roll. (note this is asked as an interview question, so resources available are limited)

My question: does the re-roll matter in terms of calculating the answer? Some people seem to think that: "Then, part of the trick is realizing that the second roll doesn't matter. Whatever the strategy is for the second roll, both parties will use it and therefore, their chances of winnings are the same. "

If that is the case, I can understand a way to get to the solution. Let ##W## represent the event that the person with the 20-sided die wins the game. Then,

$$ P(W) = P(W|\text{20-side die} \leq 12)\cdot P(\text{20-side die} \leq 12) + P(W|\text{20-side die} \geq 13)\cdot P(13 \leq \text{20-side die} \leq 20) $$

where ## P(W|\text{20-side die} \geq 13) = 1 ## and ## P(W|\text{20-side die} \leq 12) = \frac{\frac{144 - 12}{2}}{144} = \frac{11}{24} ## and these can lead to the answer. However, I don't get that assumption about the second roll.

Any help would be greatly appreciated.
 
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I think you should be able to do a lot more analysis than this. I don't see the argument about the second roll being irrelevant. Consider, for example, a 20-sided die against a two-sided die. The second roll is of little use to 2-sided die, but much more use to the 20-sided die, who can practically guarantee a win by re-rolling whenever he gets a 1 or 2.

So, that argument doesn't hold up in my view.

Next, the odds should be easy to calculate for one roll.

Finally, each player must choose a number below which they will re-roll. The odds will vary according to these two decisions, so I imagine we are looking for an optimum strategy for both players.

You could calculate (or simulate using Python) all the possible strategies for both players and see what you get.
 
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Master1022 said:
"Then, part of the trick is realizing that the second roll doesn't matter. Whatever the strategy is for the second roll, both parties will use it and therefore, their chances of winnings are the same. "
The better trick is not to trust opinions of random commentators online.
 
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FAQ: Probability: Dice Game between 20- and 12-sided dice with re-rolls

1. What is the probability of rolling a specific number on a 20-sided dice?

The probability of rolling a specific number on a 20-sided dice is 1/20 or 5%. This is because there are 20 possible outcomes and only one of them is the specific number you are looking for.

2. How many possible outcomes are there when rolling a 20-sided dice?

There are 20 possible outcomes when rolling a 20-sided dice. This is because there are 20 numbers on the dice and each one has an equal chance of being rolled.

3. What is the probability of rolling a higher number on a 12-sided dice than on a 20-sided dice?

The probability of rolling a higher number on a 12-sided dice than on a 20-sided dice is 7/12 or approximately 58%. This is because a 12-sided dice has a smaller range of numbers compared to a 20-sided dice, making it more likely to roll a higher number.

4. How does the probability change if we have the option to re-roll the dice?

The probability will increase if we have the option to re-roll the dice. This is because each time we re-roll, we have another chance to roll the desired number, increasing the overall probability of rolling it.

5. Is it possible to calculate the exact probability of winning this dice game?

Yes, it is possible to calculate the exact probability of winning this dice game by using mathematical formulas and the concept of probability. However, the exact probability will depend on the specific rules and conditions of the game, such as the number of re-rolls allowed and the specific numbers needed to win.

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