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Statistical probability of rolling all 18s

This discussion was created from comments split from: Ask Us Anything! (Volume 3).

ChinookUT
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Comments

  • kamuizinkamuizin Member Posts: 3,684
    alnair said:

    @nihility00 It's 3d6 for each stat, so the theoretical maximum is (3x6)x6 = 108. But you'd have VERY lucky to roll that...

    And will probally reroll before you notice it, i do this all the time TT!

    alnairSCARY_WIZARD
  • killeahkilleah Member Posts: 124
    pixie - can you add the desired amount of copies of BG:EE and computes needed to run multiple rerolls, - mod controlled to stop when the 6 x 18's pops up?

    I'm thinking a joint effort here, I can muster 4 comps myself, obviously a lot more is needed, I guess we want the time down to less than a year. :)

    lolien
  • kamuizinkamuizin Member Posts: 3,684
    @pixie359 it's not that math in fact, cos we have minimum rolls there, no matte what apparently the game doesn't roll less than 72 or something like that. Also the game tends to make 70 and 80 rolls more often, what means that other variants are in place there.

    BelgarathMTHTJ_Hookerlolien
  • pixie359pixie359 Member Posts: 251
    Yes, I should have made clear that that was just what happens if you go for 3 x d6 per stat with no limitations, and I know the game doesn't use that. I don't know about how it generates the random number, but one way to check would be to roll a hundred times and see how it falls - as @zarakinthish says, the results of 3 x d6 would be a bell curve peaking at 10.5, where a random number generator would be flat.

    Having one stat between 17-18 (Paladin) decreases the odds hugely - you would multiply the rest of the combined odds by two, rather than 216, assuming they're equal chance.

    lolien
  • pixie359pixie359 Member Posts: 251
    @killeah Well, how many rollers you want depends on how long you want it to take. It would take one computer 3.2 million years, and 3.2 million computers 1 year.

    Bear in mind this is an average, not a guarantee - it could happen first roll, or it could take 20 million years.

  • KaltzorKaltzor Member Posts: 1,050
    edited September 2013
    I suppose an alternative method of calculating it might be to take the 1/6 chance of one d6... Then add the 18 dice as an exponentiation... (1/6)^18... It should be the % chance of getting 18 6's in a row with a 18d6...

    Windows' Calculator says... 9,8464004200485121993638286900206e-15... That comes up to about... 0.00000000000001% chance of rolling it?

  • HadarHadar Member Posts: 168
    Children of Bhalla website have calculated it here: http://baldur.cob-bg.pl/bg2?pg=1,1,0 (Polish only, but the table is understandable for anybody)

    they give 0,000000000018413% to roll 108 points...

  • MathsorcererMathsorcerer Member Posts: 3,009
    p(rolling an 18 on 3d6) = 0.00462963 or 0.017857143, depending upon whether you are using 1/216 (1/6^3) or 1/56 (reduced results, where you count a result of 1/2/4 the same as 1/4/2--the order of results don't matter, only the sum). The first method would give p(rolling all 18s) = (1/216)^6 = 9.8464*10^-15 and the second method gives p(rolling all 18s) = (1/56)^6 = 3.24244*10^-11. Either way, the probability is so small that we can essentially say that it will never happen.

    lolien
  • SchmetterlingSchmetterling Member Posts: 5
    edited September 2013
    Hadar said:

    Children of Bhalla website have calculated it here: http://baldur.cob-bg.pl/bg2?pg=1,1,0 (Polish only, but the table is understandable for anybody)

    they give 0,000000000018413% to roll 108 points...

    Just so I understand, the column on the right shows the chance to roll *at least* the value on the left, correct? I settled on 88 (being a first-time filthy casual on my first playthrough) after much re-rolling, but even a 1/500 chance seems too generous for how long I clicked that reroll button for...

  • AstroBryGuyAstroBryGuy Member Posts: 3,415

    As great as it is that you were able to make such a statistical analysis @pixie359, as @kamuizin has pointed out, your odds are quite a bit off. First off, I believe a simplified random number generator is used by the game, which means that instead of generating three numbers from one to six and adding them together, it simply generates a number from three to eighteen. The difference being that the complex method results in a slight bell curve and the simplified version is a line. Second, since the method of character creation differs from pen and paper (i.e. you choose a class then roll, instead of rolling then figuring out what you can be), your odds of getting all 18s can be vastly improved depending on what class you choose (paladin being most likely, followed by ranger, and bard coming in third).

    Whether or not your class affects the odds of all 18s (or all maxxed, in the case of races that can get 19s) depends on how the game determines scores.

    If the game uses a standard number generation method for every score from 3-18, checks the scores against class/race minimums, and raises them to the minumum if necessary, then the odds of "rolling" an 18 are unchanged (i.e., before you re-assign points). In this method, a paladin would get a 17 Charisma like 95%+ of the time, since every generated score of 16 or below would get bumped to 17, but he'd still have to "roll" an 18 to have 18 Charisma (before you reassign points). So, to get all 18s, you'd still have to have the random number generator generate six "raw 18s".

    If the game uses a mechanic that first looks at the allowed range for each score (based on class & race), and then generates a score based on a certain min and max, then yes, the paladin's high minimum Charisma increases the odds of "rolling" an 18.

    Grammarsalad
  • HadarHadar Member Posts: 168
    edited September 2013

    Hadar said:

    Children of Bhalla website have calculated it here: http://baldur.cob-bg.pl/bg2?pg=1,1,0 (Polish only, but the table is understandable for anybody)

    they give 0,000000000018413% to roll 108 points...

    Just so I understand, the column on the right shows the chance to roll *at least* the value on the left, correct? I settled on 88 (being a first-time filthy casual on my first playthrough) after much re-rolling, but even a 1/500 chance seems too generous for how long I clicked that reroll button for...


    the column on the right shows the chance to roll *exactly* the value on the left, and its modelled for a mage (who has only one minimum score - INT must 9 or higher), paladin has usually 3 points more than the table shows (Paladin has 75% to have CHA17 and 25% to have CHA18)

  • JarrakulJarrakul Member Posts: 2,029

    p(rolling an 18 on 3d6) = 0.00462963 or 0.017857143, depending upon whether you are using 1/216 (1/6^3) or 1/56 (reduced results, where you count a result of 1/2/4 the same as 1/4/2--the order of results don't matter, only the sum). The first method would give p(rolling all 18s) = (1/216)^6 = 9.8464*10^-15 and the second method gives p(rolling all 18s) = (1/56)^6 = 3.24244*10^-11. Either way, the probability is so small that we can essentially say that it will never happen.

    That's... totally untrue. First off, outside of Bayesian probabilities with different priors (which isn't really what we're working with here, awesome though it may be), you can't get two different probabilities for the same event under the same circumstances. Statistics does not work that way. There is only one probability of rolling an 18 on 3d6, and that is 1/216. This is true because, in the cases of maximal and minimal results, there is only one possible combination that leads to the result you're looking for. If we were instead counting, say, 10s, there'd be a lot of ways to get our target result (hence the bell curve probability). Notably, no matter what we roll on the first die, it's still possible to hit 10 depending on what you roll on the other two. This allows for multiple rolls to get the same total, increasing the total probability far above the 1/216 chance for each unique ordered roll. This is not true when our target is 18, because there is only one roll that gets us there. If we roll anything but a 6 on the first die, or the second, or the third, we're done. 6/6/6 is the only roll that we can possibly make that results in 18. Because all the numbers are the same, they can't be switched around. In statistical terms, the number we're looking for on each die is independent of the number we rolled on the last. Since that means the three probabilities don't affect each other, the total probability can only be the product of the three probabilities 1/6. Hence, the result is 1/216.

    Obviously, figuring out the actual probability in the game is somewhat complicated, given uncertainties in the RNG, different racial/class minimums, and the fact that the game won't ever roll below 75 total. If we figure out precisely how the RNG works and define a race and class we can theoretically compute a probability of rolling all 18s, or if we don't want it to be a pain in the ass we can write a computer program to brute force most of it for us.

    FinneousPJTJ_HookerlolienGotural
  • the_spyderthe_spyder Member Posts: 5,018
    When I was young and played PnP, we had this one kid join the group. My DM was very strict about rolling up stats and so when the kid presented his Paladin with all 18s the DM called "Bull". The kid proudly proclaimed that he had in fact rolled all 18s (oh, and a 00 for strength on top of everything else). When asked to prove it, the kid pulled out this computer program that "randomly" generated stats for characters. after rolling some ridiculous number of times, eventually the program spit out the stats.

    Needless to say the DM disallowed the character. The DM told me later that he almost allowed the character but fully intended to infect him with stat reducing diseases that magically bypassed the Paladin's immunity, starting with Charisma. My DM could really be a piece of work sometimes. I thought it was awesome.

    FredjoGrammarsalad
  • FinneousPJFinneousPJ Member Posts: 6,456
  • MathsorcererMathsorcerer Member Posts: 3,009
    Jarrakul said:


    That's... totally untrue. First off, outside of Bayesian probabilities with different priors (which isn't really what we're working with here, awesome though it may be), you can't get two different probabilities for the same event under the same circumstances. Statistics does not work that way. There is only one probability of rolling an 18 on 3d6, and that is 1/216.

    From a textbook viewpoint you are completely correct--when you roll 3d6 there are 216 possible results and an 18, being all 6s, can occur in only one way. You are also correct that dice throws are not Bayesian.
    If, on the other hand, you are concerned only about the *sum total* of the dice, not the individual die results, then you will find that the possible outcomes collapse from 216 into only 56. Let me expand on my earlier example. Suppose you roll a 2, a 3, and a 4, giving you a result of 8. You could have rolled 2, 4, and 3 to arrive at the same result. Since we care only about the sum and not the individual results, the rolls of 2/3/4 and 2/4/3 are the same. When you examine the results this way, what you will find is that the results are as follows:
    3 has 1 outcome, 4 has only 1, 5 has 2, 6 has 3, 7 has 4, 8 has 5, each of 9 through 12 has 6, 13 has 5, 14 has 4, 15 has 3, 16 has 2, and 17 and 18 both have only 1, for a total of 56.
    Following this analysis, p(rolling an 18) = 1/56. Given that the end result is our goal rather than the individual rolls, this is a more accurate method, *in my opinion*.

    You may continue to disagree with me if you so desire but that does not invalidate my results.

    @Mathsorcerer Your username makes me sad.

    Fortunately for you, my username is irrelevant and thus you need not waste any mental energy worrying about it.

    lolien
  • HadarHadar Member Posts: 168
    edited September 2013
    @Mathsorcerer

    But you do not roll a dice with 56 faces on each with sum of the numbers but you roll six times 3 dices with 6 faces. There is no difference if you roll 18 times dice with 6 faces altogether or each separately or if you roll six times 3 dices with 6 faces. The probability of 18 in some stat is 1/216. The probability of all max stats is 1/(6^18). But due to the fact that you cannot have a PC with stats like 3/3/3/3/3/3 the probability of 18/18/18/18/18/18 is somewhat higher that 1/(6^18) (the table here http://baldur.cob-bg.pl/bg2?pg=1,1,0 is assumed to a character minimum is 3/3/3/3/9/3 (mage) - if higher are the minimum stats the easier is to get very high stats)...

    Gotural
  • pixie359pixie359 Member Posts: 251
    @mathsorcerer how about if you had a red dice, a green dice and a blue dice? If they are identifiable, you can see it has to be 216, not 56, right? Does that demonstrate that the dice aren't interchangeable when working out the odds, or do you think that the odds are different depending on the colour?

  • MathsorcererMathsorcerer Member Posts: 3,009
    edited September 2013
    I didn't say rolling dice with 56 faces; I said that there are 56 possible results if you care only about the sum and not the individual rolls. If you still doubt the results then look into it for yourself--open a spreadsheet, list all 216 outcomes of 3d6, note the sum of the 3 rolls, then count how many results are a 3, a 4, etc. and let me know what you come up with. It takes a little time, I know, but to save time you can double-check 2d6. Calculating the results this way, focusing only on the sum and not the individual rolls, will show you that rather than 36 possible rolls there are only 21: 2/2/11/12 all have only 1, 4/5/9/10 have 2, and 6/7/8/ have 3 results each.
    pixie359 said:

    @mathsorcerer how about if you had a red dice, a green dice and a blue dice? If they are identifiable, you can see it has to be 216, not 56, right? Does that demonstrate that the dice aren't interchangeable when working out the odds, or do you think that the odds are different depending on the colour?

    Ah so....that is a different problem altogether. In this example, yes you would be concerned with the individual rolls. In the generic example originally asked we do not care about the individual rolls, only the sum total of the roll itself.

  • pixie359pixie359 Member Posts: 251
    The colour of the dice has no impact on the dice and probabilities, only your experience of them.

    TJ_Hooker
  • MathsorcererMathsorcerer Member Posts: 3,009
    The real question boils down to this: do you differentiate between a roll of 2/3/4 and a roll of 4/2/3 or do you consider those to the be the same thing? If you are of the opinion that they are different then there are 216 results from rolling 3d6; however, if you are of the opinion that they are the same then there are only 56 results when you roll 3d6. Probability textbooks will tell you--correctly--that there are 216 results and that is what you should use when working problems from that book. The realistic results, where only the sum matters, is not textbook and this is the approach I am using.

  • pixie359pixie359 Member Posts: 251
    No. That is not the realistic result, and is a useless way to work things out. Roll 3 dice 1000 times and tell me how times you get 18. On average it's going to be more like 5 times than 20. Because maths.

    TJ_HookerGotural
  • pixie359pixie359 Member Posts: 251
    Another angle - there is one way of rolling 18, but 3 ways of rolling 17. There is no meaningful difference in practice between 5/6/6, 6/5/6 and 6/6/5. That they are all equal doesn't make it equally likely as 18 - it is clearly 3x more likely. No?

    alnairTJ_HookerlolienGotural
  • HadarHadar Member Posts: 168
    @Mathsorcerer

    are you studying math (I hope not :E) or the math in your nick is just for camouflage? Take 2 dices with 6 faces and roll about 100 times (if you roll a 1000 times it would be better), note the scores and then look how many times you rolled sum 2, 12, 5 and 7

    I say that you will have
    2 about 2,8% times
    12 about 2,8% times
    5 about 11,1% times
    7 about 16,7% times

    You say you will have
    2 about 4,8% times
    12 about 4,8% times
    5 about 9,5% times
    7 about 14,3% times

    Then come back and say who was closer to truth...

  • kamuizinkamuizin Member Posts: 3,684
    would the dice be influenced by any flaw in it's design? Would the initial position of whomever throw the dice influence the result? Will the wind on the place influence the results? We're kinda of walking straight into chaos mathematic here? Don't know if this kind of discussion will be productive, but then, it's on topic, so carry on!

  • CorvinoCorvino Member Posts: 2,269
    edited September 2013
    This thread appears to have rather derailed. There is only one way to roll an 18 and that is 6/6/6. I'm sure praticing christians may disapprove, number of the beast and all that. A one in 216 chance, regardless of combination of independent variables or whatever. The total number of combinations is irrelevant in this circumstance.

    alnairTJ_HookerAstroBryGuylolien
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