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  • #81767
    Mike
    Keymaster

    Which is more likely:

    Rolling 3d6 with at least one 6 in there and the total being greater than a roll of 2d6.

    Rolling 3d6 and it being at least twice as high as a roll of 2d6.

    And can you assign some sort of % so I can determine how likely each is please?
    Hopefully that makes sense?

    #81770
    Andrew Rolph
    Participant

    Mike

    Rolling 3d6 with at least one 6 in there and the total being greater than a roll of 2d6.

    There are 91 ways (out of 216 possible combinations) of rolling 3D6 such that at least one six is rolled. The lowest number it is possible to roll in these 91 instances is eight. The average number rolled on 2D6 is seven. There are 21 from 36 combinations which add to seven or less and which therefore will always lose against any 3D6 roll which includes a six. Beyond that, any roll of 13 or greater on 3D6 including a six, will also automatically beat anything rolled on 2D6 (46 of the 91 possibilities). In the end then, if you roll a six on your 3D6, the majority of your rolls will beat the 2D6 (just over 85 of the 91 possibilities, on average). These 85ish ‘winning’ rolls, as a proportion of the 216 rolls of 3D6 you could make, constitute around a 39.43% chance of beating 2D6.

    Rolling 3d6 and it being at least twice as high as a roll of 2d6.

    This is a more straightforward comparison of the range of numbers each combination of dice can roll and their respective probabilities. 3D6 produces a range of 3 to 18. 3 will never beat and roll of 2D6 doubled, 4 will be a 2D6 roll of 2 only, as will five. This pattern continues with two steps on the 3D6 roll being required to outstrip the doubled 2D6 until 18 eventually beats a roll of nine. Comparing and combining the probability of rolling any given number on 3D6 with rolling a number half as large or less on 2D6 results in 3D6 winning 30.59% of the time (just over 68 of the 216 possible rolls of 3D6).

     

    Cheers

    Andrew

    #81874
    Phil Dutré
    Participant

    Rolling 3d6 with at least one 6 in there and the total being greater than a roll of 2d6.

    Do you mean:

    “Rolling higher on 3D6, given that there is a 6” (6 is a pre-condition)
    or

    “Rolling higher on 3D6 and there must be a 6 in the 3D6 roll” (6 is a post-condition)

    Tiny Tin Men Blog: http://snv-ttm.blogspot.com/
    Wargaming Mechanics Blog: http://wargaming-mechanics.blogspot.com/

    #81878
    Phil Dutré
    Participant

    results in 3D6 winning 30.59% of the time

    Is it possible you have included the ties as well?

    Tiny Tin Men Blog: http://snv-ttm.blogspot.com/
    Wargaming Mechanics Blog: http://wargaming-mechanics.blogspot.com/

    #81910
    Phil Dutré
    Participant

    BTW, http://www.anydice.com is a great tool to answer questions such as these.

    1. Go to anydice.com
    2. type: output 3d6 – 2*(2d6)
    3. select “at least”
    4. read out the result for the value 1 in any of the various output formats (you want the value 1, because if 3d6 > 2*(2d6), that means 3d6 – 2*(2d6) >= 1

    A trick:

    When comparing dice distributions like this, you can also work with the complementary distribution. The distribution of a single d6 is the same as the distribution of 7-d6 (see also my blogpost http://wargaming-mechanics.blogspot.be/2018/01/whats-probability-xd6-beats-yd6.html).
    Thus:
    probability (3d6 – 2*(2d6) > 0)
    = probability (3d6 – 2 *(14-2d6) > 0)
    = probability (3d6+2*(2d6) > 28)

    Again you can compute this using http://www.anydice.com, by asking for the distribution of 3d6+2*(2d6)

    • This reply was modified 3 years, 5 months ago by Phil Dutré.

    Tiny Tin Men Blog: http://snv-ttm.blogspot.com/
    Wargaming Mechanics Blog: http://wargaming-mechanics.blogspot.com/

    #81939
    Andrew Rolph
    Participant

    results in 3D6 winning 30.59% of the time

    Is it possible you have included the ties as well?

     

    I don’t believe so…but I am not going to claim infallibility. If so that would merely strengthen the case for the ‘3D6 with a 6’ having the greater chance of success of the two options.

    Cheers

    Andrew

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