Who's feeling mathy? Odds of not rolling over 15 at least once.

[Dalebert]

So with standard point buy, you can't get higher than a(n expensive 15). If I have the option to roll, should I? I'm actually not bad at probability but this one's a little tedious and I'm drunk right now. Who gets excited about numbers and wants to help me figure this one out?

It's a big deal when you think about it. Score a 16 and get a +2 from race and you're basically getting a free feat. Score a natural 17 and your first feat could be one that gives +1 in a stat. I suppose if you roll a natural 15, you're likely not any worse off. The point buy seems pretty conservative compared to 4d6, drop the lowest, but that's just my gut and from rolling up a bunch of stats and seeing the trend. That's not good math/science.

I believe the way to figure it is to figure the odds of rolling 16+ once out of six time adn then taking the inverse (1 - odds). Yes? But, ugh. Lots of maths for my intoxicated brain!

[hawklost]

Simple, we will use our favorite internet program AnyDice!

For a 3d6
18 is .46%
17 is 1.39%
16 is 2.78%

So there is a 4.63% to get a 16 or higher

For a 4d6 (Drop lowest)
18 is 1.62
17 is 4.17
16 is 7.25

So there is a 13.03% chance to get 16 or higher (worth it considering)

So you can see the math, here it is
http://anydice.com/program/552c

[Symphony]

Simple, we will use our favorite internet program AnyDice!

For a 3d6
18 is .46%
17 is 1.39%
16 is 2.78%

So there is a 4.63% to get a 16 or higher

For a 4d6 (Drop lowest)
18 is 1.62
17 is 4.17
16 is 7.25

So there is a 13.03% chance to get 16 or higher (worth it considering)

So you can see the math, here it is
http://anydice.com/program/552c

Unless I'm misremembering my statistics, if it's 13.03% chance to get 16 or higher on a single 4d6 (drop lowest), then there is a 1 - (1-0.1303)^n chance of getting a 16 or higher in n rolls, which is a 56.72% chance in six rolls.

However, there is also a 29.73% chance that you will get at least one roll totaling 7 or less.

[Knaight]

There's a 13.03 percent chance to get a 16+ on any given roll with 4d6 drop highest. That leaves an 86.97% chance of not getting it on any given roll, which gives us 1-((.8697)^6) for the chance of getting at least one 16. That's .5672, or a 56.72% chance of getting at least one 16. Or, to directly answer the thread question, the odds of not rolling over 15 at least once are 56.72 to 43.28, or 1.31 to 1. Going with percentages, that's a 43.28% of not rolling over 15 at least once.

[xyianth]

Now that the math has been answered, here is a semi-joking answer. If your DM allows rolling, you should roll. If you get lucky, it can provide a great boost. If you get unlucky, you have two options: Die quickly and roll a new character (new chance to be lucky on stats), or use with the worse stats and realize that missing a +1 or +2 won't make you useless thanks to bounded accuracy. (every +1 is just 5% better, starting with a 12 in your main stat is only a 15% reduction from starting with an 18)

Personally, I prefer point buy games simply because games seem to flow better when all members of the party are on a level playing field. Rolling always seems to give at least one player a lucky advantage that the other players feel like they have to catch up to.

[Dalebert]

There's a 13.03 percent chance to get a 16+ on any given roll with 4d6 drop highest. That leaves an 86.97% chance of not getting it on any given roll, which gives us 1-((.8697)^6) for the chance of getting at least one 16. That's .5672, or a 56.72% chance of getting at least one 16. Or, to directly answer the thread question, the odds of not rolling over 15 at least once are 56.72 to 43.28, or 1.31 to 1. Going with percentages, that's a 43.28% of not rolling over 15 at least once.

Your math seems accurate but it doesn't seem in line with my actual rolls. Maybe I just got a lot of lucky rolls, but rolling stats about 12 times, I got about 75% with at least one 16+ and often more. Hmm... Quite possibly just not a big enough sampling to reflect the odds better. I'm still inclined to go with random. Even if I get just one 15 and a few above averages, that's somewhat comparable to point buy. In fact, I should really be comparing to the odds of not getting at least one 15+. That would make me clearly worse off than if I did straight point buy.

[hawklost]

Your math seems accurate but it doesn't seem in line with my actual rolls. Maybe I just got a lot of lucky rolls, but rolling stats about 12 times, I got about 75% with at least one 16+ and often more. Hmm... Quite possibly just not a big enough sampling to reflect the odds better. I'm still inclined to go with random. Even if I get just one 15 and a few above averages, that's somewhat comparable to point buy.

I will point out that math and what we do with a very small subset are not going to be the same.

Math calculates to the effective infinity.

Your roll consisted of 12 attempted.

Its like saying that flipping a coin have a 50% chance of landing on heads and then someone testing this 10 times and having 7 heads, it is still 50% but your subset was too small to get that close.

[Dalebert]

If your DM allows rolling, you should roll.

I'm very inclined to agree. Also, this DM is a fellow player in another game and was allowed to roll and got very lucky. He also encourages rolling for hit points quite a bit. He makes us round down if we take the average and gives us +1 if we roll! Of course I roll HP in those conditions (and my first roll was a 1, of course). I'll be very surprised if he doesn't let us roll stats. I'm actually shooting for some kind of encouragement to roll but I'll roll even if he doesn't provide it.


Your roll consisted of 12 attempted.

Its like saying that flipping a coin have a 50% chance of landing on heads and then someone testing this 10 times and having 7 heads, it is still 50% but your subset was too small to get that close.

Yes, conceded. That's essentially what I recently said--


Quite possibly just not a big enough sampling to reflect the odds better.

[Knaight]

I will point out that math and what we do with a very small subset are not going to be the same.

Math calculates to the effective infinity.

The equations in use here calculate averages that will reliably be approached and are guaranteed to be at effective infinity. It's not like there isn't math that looks at the small subset - pretty basic binomial probability could be employed to see how likely that 9/12 stat generations having at least 1 16 was. Even from a numerical intuition standpoint, 6.81 sets of stats with at least 1 16 out of 12 are expected, 9 is reasonably close and still decently far from the extremes of the distribution, and thus getting 9 sets with at least 1 16 is decently likely.

I'd calculate the actual odds, but getting them from 9-12 is on the tedious end without a program or calculator function dedicated to it, and I've got typing equations into Google or Wolfram Alpha here.

[woodlandkammao]

Here's an even more detailed one that includes the odds of each of your six rolls being a given number.
http://catlikecoding.com/blog/post:4d6_drop_lowest

[woodlandkammao]

also, if you're rerolling 1's, here's anydice odds for that.

http://anydice.com/program/5532

[Rilak]

Average point buy values (assuming you value 16,17,18 at 3pt increase each, and 3-7 at 0.5pt penalty):

4d6, drop lowest
3 0.077% -0.002 ( -2.5)
4 0.309% -0.006 ( -2.0)
5 0.772% -0.012 ( -1.5)
6 1.620% -0.016 ( -1.0)
7 2.932% -0.015 ( -0.5)
8 4.784% 0.000 ( +0.0)
9 7.022% 0.070 ( +1.0)
10 9.414% 0.188 ( +2.0)
11 11.420% 0.343 ( +3.0)
12 12.886% 0.515 ( +4.0)
13 13.272% 0.664 ( +5.0)
14 12.346% 0.864 ( +7.0)
15 10.108% 0.910 ( +9.0)
16 7.253% 0.870 (+12.0)
17 4.167% 0.625 (+15.0)
18 1.620% 0.292 (+18.0)
Total 100.000% 5.291
Total, 6 rolls 31.743 point buy

4d6, drop lowest, reroll 1s once
3 0.000% -0.000 ( -2.5)
4 0.002% -0.000 ( -2.0)
5 0.019% -0.000 ( -1.5)
6 0.261% -0.003 ( -1.0)
7 0.871% -0.004 ( -0.5)
8 1.991% 0.000 ( +0.0)
9 3.906% 0.039 ( +1.0)
10 6.727% 0.135 ( +2.0)
11 9.814% 0.294 ( +3.0)
12 12.880% 0.515 ( +4.0)
13 14.925% 0.746 ( +5.0)
14 15.520% 1.086 ( +7.0)
15 13.825% 1.244 ( +9.0)
16 10.497% 1.260 (+12.0)
17 6.249% 0.937 (+15.0)
18 2.512% 0.452 (+18.0)
Total 100.000% 6.702
Total, 6 rolls 40.212 point buy

3d6
3 0.463% -0.012 ( -2.5)
4 1.389% -0.028 ( -2.0)
5 2.778% -0.042 ( -1.5)
6 4.630% -0.046 ( -1.0)
7 6.944% -0.035 ( -0.5)
8 9.722% 0.000 ( +0.0)
9 11.574% 0.116 ( +1.0)
10 12.500% 0.250 ( +2.0)
11 12.500% 0.375 ( +3.0)
12 11.574% 0.463 ( +4.0)
13 9.722% 0.486 ( +5.0)
14 6.944% 0.486 ( +7.0)
15 4.630% 0.417 ( +9.0)
16 2.778% 0.333 (+12.0)
17 1.389% 0.208 (+15.0)
18 0.463% 0.083 (+18.0)
Total 100.000% 3.056
Total, 6 rolls 18.333 point buy


So you get on average better results by using 4d6, and you may end up with better than the best possible stat in point buy at a reasonable rate.

[Chronos]

Some DMs houserule that you can roll, and if your rolls are below average, you can take average instead, or something like that. In that case, you'd be a fool not to roll. Short of a houserule like that, though, you shouldn't take it. Yes, you might get really good rolls... but you might get really bad rolls, too. Every character concept is playable with the normal array, or especially with point buy, but not every character concept is playable with every possible roll of stats. So you're basically trading a guarantee that you can play what you want for a chance that you can play what you want.

[Kyutaru]

(every +1 is just 5% better, starting with a 12 in your main stat is only a 15% reduction from starting with an 18)

Not exactly, it's 5% out of 20, but 20 is an irrelevant value in combat. It's just the range of the values.

Yet if you built your fighter so that he needs a 16 to hit a certain mob, that's a 25% chance of success. A +1 to hit means a 15 or higher hits, improving your chance to 30%. While 30% is indeed 5% separation out of the whole range of numbers you can roll, it's a 20% improvement on your chances. Let's take the opposite, let's say an enemy needs a 19 or higher to hit your insanely high AC. That's a 10% chance of success. A +1 to your AC would make this a 20 or higher, or a 5% chance of success, or effectively halving his chances (-50%), or doubling your survivability (+100%).

Suddenly those +1s seem to make a very powerful difference even with bounded accuracy. This wasn't as much an issue in 3rd edition where you were extremely likely to hit the target if you min-maxed. Bounded accuracy takes us away from the autopass-autofail mechanic, but it doesn't trivialize a +1 bonus. If anything that +1 sword is even more valuable now, and having good stats can make you overwhelmingly stronger.

[Chronos]

And the value of each +1 increases even more once you have either advantage or disadvantage, or mechanics like "roll saves until you get three successes or three failures".

[xyianth]

Not exactly, it's 5% out of 20, but 20 is an irrelevant value in combat. It's just the range of the values.

Yet if you built your fighter so that he needs a 16 to hit a certain mob, that's a 25% chance of success. A +1 to hit means a 15 or higher hits, improving your chance to 30%. While 30% is indeed 5% separation out of the whole range of numbers you can roll, it's a 20% improvement on your chances. Let's take the opposite, let's say an enemy needs a 19 or higher to hit your insanely high AC. That's a 10% chance of success. A +1 to your AC would make this a 20 or higher, or a 5% chance of success, or effectively halving his chances (-50%), or doubling your survivability (+100%).

Suddenly those +1s seem to make a very powerful difference even with bounded accuracy. This wasn't as much an issue in 3rd edition where you were extremely likely to hit the target if you min-maxed. Bounded accuracy takes us away from the autopass-autofail mechanic, but it doesn't trivialize a +1 bonus. If anything that +1 sword is even more valuable now, and having good stats can make you overwhelmingly stronger.

Only if you are fighting things at the extreme range of possible. The entire point of bounded accuracy was to remove situations like that. If you need a 16 to hit a monster, you shouldn't be fighting that monster at your level. (or you are fighting some legendary behemoth with ridiculous stats) Assuming you have a 12 in your main stat, (unlikely unless you have terrible rolls or you are being intentionally bad with your ASIs) needing a 16 to hit means you are fighting things with 19 AC from 1-4, 20 AC from 5-8, 21 AC from 9-12, 22 AC from 13-16, and 23 AC from 17+. Look through the MM for CR appropriate foes with those numbers. They are very rare.

I have personally witnessed the effect of having lowish stats has in 5e, in the vast majority of cases you miss maybe one more time per combat than your friends. One of the things I do every edition is try and build a character with 8s in every stat and see if any part of the game breaks. (in 3.5, I did it with all 3s) 5e (and 3.5) do not require max stats and perfect +1s to be successful. 4e did because the whole game was a massive treadmill with optimization assumed and a target miss chance of 50%. And 1e+2e killed you fast even with great stats. You absolutely don't need to sweat the difference between starting with a 14 vs starting with a 16 or 18 in your main stat in 5e.

[Knaight]

Only if you are fighting things at the extreme range of possible. The entire point of bounded accuracy was to remove situations like that. If you need a 16 to hit a monster, you shouldn't be fighting that monster at your level. (or you are fighting some legendary behemoth with ridiculous stats) Assuming you have a 12 in your main stat, (unlikely unless you have terrible rolls or you are being intentionally bad with your ASIs) needing a 16 to hit means you are fighting things with 19 AC from 1-4, 20 AC from 5-8, 21 AC from 9-12, 22 AC from 13-16, and 23 AC from 17+. Look through the MM for CR appropriate foes with those numbers. They are very rare.

A 16 isn't out in the extreme at all, look at skill checks (where one might have +4 on a trained skill for a secondary attribute and need a 20 for something). D&D 5e is designed to mostly keep hit chances within the center of the range, but at the very least that encompasses 6-14. Plus, going from 11 needed to 10 needed is a 10% increase in expected damage output, which is still well over the 5% stated.

[Easy_Lee]

Sources of +1 are comparatively hard to come by this edition, due to a general weakening of PCs, widespread use of advantage and disadvantage instead of pluses and minuses, and fewer stacking bonuses from less common magic items. With all of this together, +1 from stats is very important. So I would roll stats if given the option and start with my main stats as high as possible.

That said, if the DM let me point buy a 16 and a 15, I would go with that over rolling.

[xyianth]

Ok, I think we are arguing past each other a bit. Let me clarify my point.

This thread is about whether a player, given the choice, should roll for stats or use the point buy system.
My answer is: the player should roll.

This is my reasoning: rolling gives a chance at above average stats, average stats, and below average stats. I think we can all agree that above average stats are awesome. Hopefully we can all also agree that average stats means we lost very little by choosing to roll. That leaves the below average case. I am claiming that, even with below average stats, your character is playable and can be fun.

I am not saying that every +1 isn't valuable. I am saying that we (members of these forums) tend to make a bigger deal out of being optimum than we should when giving advice on characters meant to be played in actual games.

5e is not 4e. Missing a few possible +1s will not cripple your character against level appropriate foes. Having suboptimal stats will not cause you to fear normal skill challenges. Having weaker stats will not make saves any less scary than they are for optimized characters.

Can you contrive examples where a missing +1 will have a significant impact? Sure. But they are contrived examples. It is not normal to need a 16 to hit a level appropriate enemy. DC 20 checks are meant to be very difficult, even to those trained in the skill. A DC 20 save is meant to be failed most of the time by most characters.

For high AC enemies, good tactics, teamwork, and creativity are how you are supposed to deal with the threat. Can an optimized character deal with it easier? Of course, but that doesn't mean it is impossible for the non-optimized character. For high DC checks, work together, use guidance, or find a way to bypass the check. No amount of optimization will create a character that can pass DC 20 checks of every type with ease. For high DC saves, planning and preparation will overcome the challenge. This is true for optimized characters as well, since only monks and paladins will have a good chance of passing high DC saves of all abilities.

As for equating the +1 to a 5% difference, we are simply looking at it differently. I'm viewing it from the standpoint of: Out of 20 attempts at a given thing, the +1 will help me succeed on 5% more rolls. You appear to be viewing it as: On a given attempt the +1 improves my chance of success by X%. Both viewpoints are completely valid, they just represent different perspectives of the same thing. My apologies for any confusion the 5% figure may have caused.

TL;DR: roll for stats, the 'risk' of below average stats is not that big of a 'risk' in 5e.

[Chronos]

This is my reasoning: rolling gives a chance at above average stats, average stats, and below average stats. I think we can all agree that above average stats are awesome. Hopefully we can all also agree that average stats means we lost very little by choosing to roll. That leaves the below average case. I am claiming that, even with below average stats, your character is playable and can be fun.
You can't have it both ways. If your stats being a little above average is a significant benefit over average, then them being a little below average is a significant drawback. Either that +1 is significant, or it's not. Except, of course, for the full comparison you also have to look at the possibility that, instead of being a little below average, you'll be a lot below average. A character whose highest stat is a 14 instead of a 15 might still be playable, but how about a character whose highest stat is 11? That's not likely, maybe, but it can happen. Such a character would, in principle, be playable, but they're going to suck pretty badly. And you shouldn't roll for scores unless you're willing to accept that that can happen.

[xyianth]

You can't have it both ways. If your stats being a little above average is a significant benefit over average, then them being a little below average is a significant drawback. Either that +1 is significant, or it's not. Except, of course, for the full comparison you also have to look at the possibility that, instead of being a little below average, you'll be a lot below average. A character whose highest stat is a 14 instead of a 15 might still be playable, but how about a character whose highest stat is 11? That's not likely, maybe, but it can happen. Such a character would, in principle, be playable, but they're going to suck pretty badly. And you shouldn't roll for scores unless you're willing to accept that that can happen.

I said they were awesome, not significant. The way 5e handles stats means the true effect of high stats isn't really measured by being effective, its measured in how many ASIs you can spend on secondary stats or feats. (or afford to skip/delay by multiclassing)

Since your example of highest roll is 11, I'll assume you meant for that to be the characters main stat. Can I at least assume that you can choose race/class/tactic after seeing what stats you have to work with? I'm sure that rolling terribly and picking a MAD class/tactic and a race that doesn't help would be possible, but at that point you aren't really trying to be effective.


Start with an 11 dex pre-racial. Pick a race that boosts dexterity by at least 1: gives you a starting score of 12-13. Choose non-champion fighter. Select archery fighting style. Only use ASis on every 4th level to boost dex. (ignore bonus ASIs from fighter)

From levels 1-3, you have an attack mod of +5 vs AC 13 foes: you hit 65% of the time.
At level 4, you get a +2 dex bonus so your attack mod becomes +6 vs AC 14 foes: you still hit 65% of the time.
From levels 5-7, you have an attack mod of +7 vs AC 15 foes: you still hit 65% of the time.
At level 8, you get another +2 dex bonus so your attack mod becomes +8 vs AC 16 foes: you still hit 65% of the time.
At level 9, you have an attack mod of +9 vs AC 16 foes: you now hit 70% of the time.
From levels 10-11, you have an attack mod of +9 vs AC 17 foes: you now hit 65% of the time.
At level 12, you get another +2 dex bonus so your attack mod becomes +10 vs AC 17 foes: you now hit 70% of the time.
From levels 13-15, you have an attack mod of +11 vs AC 18 foes: you still hit 70% of the time.
At level 16, you get another +2 dex bonus so your attack mod becomes +12 vs AC 18 foes: you now hit 75% of the time.
From levels 17-20, you have an attack mod of +13 vs AC 19 foes: you still hit 75% of the time.

Now we do the same thing for a character that started with an 18 dexterity pre-racial. Lets assume a wood elf for the +2 dex and archery style just to make the comparison fair:

From levels 1-3, you have an attack mod of +9 vs AC 13 foes: you hit 85% of the time.
At level 4, your attack mod is still +9 vs AC 14 foes: you now hit 80% of the time.
From levels 5-7, you have an attack mod of +10 vs AC 15 foes: you still hit 80% of the time.
At level 8, your attack mod is still +10 vs AC 16 foes: you now hit 75% of the time.
At level 9, you have an attack mod of +11 vs AC 16 foes: you now hit 80% of the time.
From levels 10-11, you have an attack mod of +11 vs AC 17 foes: you now hit 75% of the time.
At level 12, your attack mod is still +11 vs AC 17 foes: you still hit 75% of the time.
From levels 13-15, you have an attack mod of +12 vs AC 18 foes: you still hit 75% of the time.
At level 16, your attack mod is still +12 vs AC 18 foes: you still hit 75% of the time.
From levels 17-20, you have an attack mod of +13 vs AC 19 foes: you still hit 75% of the time.
Clearly, starting with a 20 in your main stat is better than starting with a 12. But does missing 1 extra attack per 5 attempts mean you suck? I would say no. Having high stats does mean that you have more ASIs to spend on secondary stat boosts or feats. And, this assumes you rolled 6x 4d6b3 and got nothing higher than an 11. (an admittedly unlikely scenario) If you get an even less likely scenario, the math does get worse. As long as your main stat is at least a 6 post-racial, you will still be able to cap it by 20th fighter, but your hit chance does dip below 50% for the early game though. I believe this is why point buy's minimum score is an 8.

Edit: Apparently, my brain was half asleep when doing the math for the wood elf from levels 12+, I fixed the math so it's now based on reality. :smallredface: Sorry for any confusion.