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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