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Pathfinder elite array is 15, 14, 13, 12, 10 and 8.

When rolling stats with 3d6, what is the probability of getting an equal or better set of ability scores than the elite array? For a definition of better, consider the point buy cost of buying that ability score array. For small ability score values, see this question: How to extend Pathfinder point buy to scores below 7?

Context

Players are resettling an abandoned settlement, and we want to focus play in the process. We'll roll 3d6 in order stats for the residents. Players can pick their character from among those, with limits on how many characters of player character class are allowed. If the population increases, they get more sets. The population will start out small, and I am considering how feasible this plan is. To determine this it is useful to know how often elite array or better gets generated with 3d6 rolls.

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    \$\begingroup\$ You really need to provide your own criteria for the definition of 'better' here. \$\endgroup\$ – Wibbs Apr 1 '18 at 8:42
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    \$\begingroup\$ If you can't specify the details of the question, you're not asking for an answer. You're asking for a question. And what defines 'better' is inherently opinion based. As written, you could receive a half dozen answers prioritizing different stats and not come up with a 'correct' answer. \$\endgroup\$ – Ifusaso Apr 1 '18 at 12:18
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    \$\begingroup\$ @Ifusaso: "How do I extend the point-buy costs downward below 6 in a way that makes sense for actual play?" Seems like a perfectly reasonable question that makes use of RPG expertise (and thus is not on-topic at Math SE). \$\endgroup\$ – user17995 Apr 1 '18 at 13:14
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    \$\begingroup\$ Your question currently consists of two. 1) What criteria should I use to determine if an ability score array is better?, and 2) What is the likelihood that a rolled set is better than this array? The first question may be too opinion based as it is currently written (you should probably revise), and you may want to separate them into two separate questions. \$\endgroup\$ – David Coffron Apr 1 '18 at 13:47
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    \$\begingroup\$ This question is still asking answerers to provide and support their criteria. That's still too broad: voters can't be expected to compare apples to orangutans. \$\endgroup\$ – SevenSidedDie Apr 1 '18 at 17:29
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Well, here's a quick AnyDice script to plot the distribution of point buy costs for some of the various score rolling methods (Standard, Classic and Heroic) listed on this page:

\ point buy cost of a single ability score \
\ source: http://www.d20pfsrd.com/basics-ability-scores/ability-scores/ \
function: cost of SCORE:n {
  if SCORE < 7 { result: -4 } \ assume that going below score 7 gives you no more points \
  if SCORE > 18 { result: 1000000 }  \ make trying to go above 18 cost a million points \
  result: (SCORE - 6) @ { -4, -2, -1, 0, 1, 2, 3, 5, 7, 10, 13, 17 }
}

\ various score rolling methods \
output 6 d [cost of [highest 3 of 4d6]] named "standard (4d6 drop lowest)"
output 6 d [cost of 3d6] named "classic (3d6)"
output 6 d [cost of 2d6 + 6] named "heroic (2d6 + 6)"

Here's what the output looks like, in normal graph mode:

Screenshot

The elite (15,14,13,12,10,8) array, of course, costs exactly 15 points to buy, so we can use the "At Least" output mode on AnyDice and look at the entry numbered "15" in each output graph to find the probability of rolling an array of scores that would cost at least 15 points to buy:

Screenshot

Doing so, we find that the probabilities for the various methods are:

  • Standard (4d6 drop lowest): 65.84%
  • Classic (3d6): 15.95%
  • Heroic (2d6 + 6): 85.52%

Note that the script above assumes that trying to buy an ability score below 7 won't give you any more points than just buying a score of 7. While that seems like a perfectly reasonably ruling to make if a player actually wanted to do that for some reason, it does also mean that the scoring scheme considers e.g. the score arrays (15,14,13,12,12,7) and (15,14,13,12,12,3) to both be worth exactly 15 points.

As an alternative, we could change the script to assign ability scores below 7 an arbitrary cost of, say, -1000 points, effectively assuming that no player would want such a low score for any number of points. Of course that's not actually true — many players would be more than happy with a score array of, say, (18,18,18,18,18,6) — but it at least sort of models the fact that, in point buy, you straight up cannot buy a score that low.

With that modification, the script produces the following output:

Screenshot

Note that I added an arbitrary cutoff to the output at -24 points (the point value of the lowest legally buyable set of scores, i.e. 7,7,7,7,7,7) to keep the plot from extending to the left into the negative thousands. Since we're really only interested in the part of the graph at 15 points and above, the cutoff makes no difference to the actual results.

With this definition of "better" (which basically considers any array with a score below 7 to be worse than any legally buyable score array), the probabilities of rolling an array worth more than 15 points become:

  • Standard (4d6 drop lowest): 59.57%
  • Classic (3d6): 12.79%
  • Heroic (2d6 + 6): 85.52%

As expected, there's no difference for the Heroic rolling method, since you straight up cannot roll below 8 on 2d6 + 6. For the Standard method, there's a 65.84% − 59.57% = 6.27% chance of rolling an array that includes at least one score below 7, and which would be worth 15 or more points if those low scores would only count as -4 points each. For the Classic method, the same happens with a probability of 15.95% − 12.79% = 3.16%.

Of course, if you had some other scheme for assigning point buy values to scores below 7 in mind, you could easily tweak the script above to try it out. In some sense, though, the two approaches implemented above (valuing scores below 7 at either -4 or -1000 points) represent the two extremes between which any sensible valuation scheme should lie.

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