Dismiss Notice
Join Physics Forums Today!
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

  1. Feb 14, 2012 #1
    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
     
  2. jcsd
  3. Feb 14, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi Paraxis! Welcome to PF! :wink:
    Isn't that the same as
    1. Roll 5 dice.
    2. Take the sum of the three highest dice?​
     
  4. Feb 14, 2012 #3
    Re: Welcome to PF!

    ...

    I've racked my brain and can't see how it is NOT!

    Wow... Just roll 5 dice... I guess I over-thought things!

    Unless someone demonstrates that it is not the same as rolling 5 dice...

    Thanks! Much MUCH easier to calculate...

    So the distribution is:

    3 = 1 (0.01%)
    4 = 5 (0.06%)
    5 = 15 (0.17%)
    6 = 41 (0.45%)
    7 = 90 (0.99%)
    8 = 171 (1.88%)
    9 = 300 (3.31%)
    10 = 485 (5.35%)
    11 = 697 (7.68%)
    12 = 946 (10.43%)
    13 = 1163 (12.82%)
    14 = 1326 (14.62%)
    15 = 1335 (14.72%)
    16 = 1196 (13.18%)
    17 = 854 (9.41%)
    18 = 447 (4.93%)
     
  5. Feb 14, 2012 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    he he :biggrin:
     
  6. Feb 14, 2012 #5
    Re: Welcome to PF!

    That totals 9072 permutations but 65 = 7776

    I make the correct frequencies 1, 5, 15, 41, 90, 170, 296, 470, 665, 881, 1055, 1155, 1111, 935, 610, 276
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Calculating probability distribution for rolling 4 dice plus reroll lowest die
  1. Dice rolling (Replies: 2)

Loading...