- #1
Paraxis
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Hi,
I am trying to figure out what the probability distribution is for the following:
1. Roll 4 dice.
2. Take the lowest die and re-roll it.
3. Take the sum of the three highest dice.
The result will be between 3 and 18.
I know how to figure out the probability distribution for rolling 4 dice and taking the three highest, but how can we calculate the distribution when re-rolling the lowest die?
My thought process goes as follows:
When you roll the 4 dice and keep the three highest, the result from re-rolling the lowest remaining die is only kept if the roll is higher than any of the three dice that were kept.
e.g.
Say you roll a 6,4,2 and 1. Keep the 6,4 and 2 and re-roll the 1.
The new roll is only kept if and only if it exceeds 2.
Therefore there is a 2 in 6 probability for the sum to be 12 and there is a 1 in 6 probability for the sum to be each of 13, 14, 15 or 16.
I can calculate this long hand by writing up all 1296 possible combinations for 4 dice, but if you could supply me with a formula, it would be much quicker...
Thanks,
Paraxis
I am trying to figure out what the probability distribution is for the following:
1. Roll 4 dice.
2. Take the lowest die and re-roll it.
3. Take the sum of the three highest dice.
The result will be between 3 and 18.
I know how to figure out the probability distribution for rolling 4 dice and taking the three highest, but how can we calculate the distribution when re-rolling the lowest die?
My thought process goes as follows:
When you roll the 4 dice and keep the three highest, the result from re-rolling the lowest remaining die is only kept if the roll is higher than any of the three dice that were kept.
e.g.
Say you roll a 6,4,2 and 1. Keep the 6,4 and 2 and re-roll the 1.
The new roll is only kept if and only if it exceeds 2.
Therefore there is a 2 in 6 probability for the sum to be 12 and there is a 1 in 6 probability for the sum to be each of 13, 14, 15 or 16.
I can calculate this long hand by writing up all 1296 possible combinations for 4 dice, but if you could supply me with a formula, it would be much quicker...
Thanks,
Paraxis