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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

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    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

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    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
     
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