Welcome Forums The principle of inverted dice Playing with 5 inverted dice

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  • #734
    Inverted Dice
    Keymaster

    The case with 5 dice is extremely interesting, because there are exactly 20 possible results from a roll using 5 inverted dice, and the results are always between 1 and 20.

    In other words:

    Five  6-sided dice can together form one  20-sided die (with a peculiar probability distribution).

    This is exactly what we use in the game Inverted Dice. On the page with the rules you’ll find several examples of dice rolls with 5 inverted dice.

    When calculating the result of an inverted dice roll, it can be useful knowing that the sum of the numbers from 1 to 6 is 21. For instance, if you only rolled threes and fives, you know that the inverted sum is 13, since 3 plus 5 equals 8, and 21 minus 8 equals 13. It is often much easier to count that way.

    And finally, here is a probability table for rolls with 5 dice (inverted):

    Chances of getting the results 1 to 20 in one roll with 5 dice (inverted).
    As you can see, the results from 16 to 20 are hard to get.

    Links:

    Playing with 1 inverted die [1]

    Playing with 2 inverted dice [1] [2]

    Playing with 3 inverted dice [1] [2] [3]

    Playing with 4 inverted dice [1] [2] [3] [4]

    Playing with 6 inverted dice [1] [2] [3] [4] [5] [6]

    Playing with many  inverted dice

    #2538
    Kant
    Participant

    Can someone explain to me, exactly why the chance of getting 1 is 120 times bigger than the chance of getting 20?
    It seems to me like you can only get the inverted sum 1 by rolling [2][3][4][5][6] and 20 by rolling [1][1][1][1][1]? Shouldn’t the chances be the same then?

    #2587
    Inverted Dice
    Keymaster

    Well, there are 120 possible rolls where the dice show no [1]’s, and [2] [3] [4] [5] [6] is just one of them. Another is [6] [3] [4] [5] [2] … and so on.

    The first die can show [2], [3], [4], [5] or [6] (i.e. 5 possible values). Let’s say it shows [2], then the second die can show [3], [4], [5] or [6] (i.e. 4 possible values). Let’s say it shows [3], then the third die can show [4], [5] or [6] (i.e. 3 possible values). Let’s say it shows [4], then the fourth die can show [5] or [6] (i.e. 2 possible values). Let’s say it shows [5], then the fifth die must show [6] (i.e. 1 possible value) in order to achieve the inverted sum 1.

    5 times 4 times 3 times 2 times 1 is 120, hence the probability 120/7776 (where 7776 is number of ways in which five dice can land).

    When it comes to [1] [1] [1] [1] [1], there is really no other way, it can only be rolled in one way.
    1 times 1 times 1 times 1 times 1 is 1, hence the probability 1/7776.

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