Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Calculating probability distribution for rolling 4 dice plus reroll lowest die

**Physics Forums | Science Articles, Homework Help, Discussion**