Odds of perfect D&D character

[thedanster7000]

I (attempted) to calculate the odds of rolling a full-18s character to prove that a particular individual hadn't (not that it needed confirming)...

I came up with 0.0000000002%, to put that in perspective, if everyone on Earth rolled a character, 13 would get full 18s. Just putting this up here to see if I did the maths correctly or so someone can point me to the correct figure otherwise, and to see if anyone else has had anyone make the claim of full 18s.

That figure assumes one rolls 4d6 and drops the lowest.

[Lalliman]

That is correct. Or technically it's 0.000000000167, but potayto potahto.

Of course, due to how many characters must have been rolled up since the invention of D&D, it's probably happened at some point. That's the law of truly large numbers.

By the way, if your player is trying to cheat by claiming he rolled all 18s, he's kind of an idiot. At least fake something more probable.

[thedanster7000]

That is correct. Or technically it's 0.000000000167, but potayto potahto.

Of course, due to how many characters must have been rolled up since the invention of D&D, it's probably happened at some point. That's the law of truly large numbers.

By the way, if your player is trying to cheat by claiming he rolled all 18s, he's kind of an idiot. At least fake something more probable.
Yeah, I rounded it for simplicity's sake.

I think he was joking (with skillful deadpan if so), and used Point Buy in the end, but it got me thinking about the actual odds.

[JBPuffin]

To be fair, it's even more impossible with straight 3d6: .000000000001% chance in straight 3d6, no rerolls...

[Yora]

Such small fractions of a percent are hard to grasp. What does that translate to? One in a billion?

[thedanster7000]

Such small fractions of a percent are hard to grasp. What does that translate to? One in a billion?
Just over 1.5 in a billion.

[Slipperychicken]

Might as well show some math here.


Chance to roll 18 in one iteration of 3d6: (1/6)^3 = 0.0046 = 0.46%
Odds of rolling six 18s in six iterations of 3d6: [1 / (0.0046)^6] = one to 105,548,729,470,595
That's one in 105 trillion to roll all 18s in 3d6 down-the-line.


Chance to roll 18 in one iteration of 4d6b3 ("4d6 best three"): 1.62%
Odds of rolling six 18s in six iterations of 4d6b3: [1 / (0.0162)^6] = 55,323,533,773.00
That's one in 55 billion. Much better chances than on 3d6, but far from realistic.

[Cluedrew]

Let me take a crack at this calculation.

The chance of getting an 18 in a with 3d4 would be: 1/6*1/6*1/6=(1/6)^3=1/216.

Adding a fourth dice, and dropping lowest, makes things a bit more complicated. Intuitively this should raise our chances (and it does), but the simplest calculation where we let the forth dice be anything [(1/6)^3*1=1/216] doesn't help. And that is because any one of the four dice can be not 6. So we get something closer to 4*(1/6)^3. But that counts the possibility of rolling 4 sixes 4 times instead of once. So instead I will use the chance of rolling exactly 6 sixes [4*(1/6)^3*(5/6)] plus the chance of rolling 4 sixes [(1/6).

4*((1/6)^3*5/6)+(1/6)^4=21/1296=7/432

Now we need to have that happen six times in a row. Can't do that in my head. Calculator time! And my calculator doesn't have enough digits. More powerful calculator time.

117 649/6 499 837 227 778 624

Or about 1.8x10-11 or roughly 1 in 50 billion.My estimate (approximately 1 in 50 billion) is closer to Slipperychicken's.

Who says math isn't fun?

[khadgar567]

its better to try super loto than get perfect character with that odds there is solid chance that you win at least major payment from it

[Xuc Xac]

On the other hand, low odds don't mean impossible. Try this little experiment at home. Flip a coin 20 times and write down the sequence of heads and tails results that you get. The odds of getting that exact sequence are about 1 in a billion, but you just did it. Every sequence has the same chance as all heads or all tails, but we only attach importance to the ones that look unique even though they're all unique.

[Yora]

However, there is only one way to get six 18s, but there are six ways of getting five 18s and one 17. And thirty-six ways to get four 18s and two 17s. The odds to get six numbers that are all 17 or higher are vastly larger than getting each die exactly 18.

[Demidos]

On the other hand, low odds don't mean impossible. Try this little experiment at home. Flip a coin 20 times and write down the sequence of heads and tails results that you get. The odds of getting that exact sequence are about 1 in a billion, but you just did it. Every sequence has the same chance as all heads or all tails, but we only attach importance to the ones that look unique even though they're all unique.

Rightt.....I get your point, but we're also talking about picking one (all 18s) BEFORE you roll. The chances of rolling that while INTENDING to roll that are pretty much nonexistent.

Cool idea, but you have to be careful applying it.

[Slipperychicken]

I think it's also worth mentioning that not all dice are fair. A set of dice biased to favor the "6" side may increase the likelihood of all 18s somewhat, while any dice biased away from that side (for instance a die that favors 4) may decrease the probability accordingly.

That said, you would need some very strongly biased dice to make such a roll likely. And it would be relatively simple to demonstrate that such dice are not even close to fair. And not in the sense that "oh dice can't be perfectly random". Like if "fair" was in London, then these dice would have to be somewhere on Jupiter or Pluto, that's how far from fair they would have to be.

[thedanster7000]

Rightt.....I get your point, but we're also talking about picking one (all 18s) BEFORE you roll. The chances of rolling that while INTENDING to roll that are pretty much nonexistent.

Cool idea, but you have to be careful applying it.
I think he's just saying that it's possible that he might have genuinely rolled 18s.

[FreddyNoNose]

That is correct. Or technically it's 0.000000000167, but potayto potahto.

Of course, due to how many characters must have been rolled up since the invention of D&D, it's probably happened at some point. That's the law of truly large numbers.

By the way, if your player is trying to cheat by claiming he rolled all 18s, he's kind of an idiot. At least fake something more probable.

I had a much harder situation. Rolling 4d6 keep best 3 nine times for six stats and my highest roll was a 5. Had several threes.

[Slipperychicken]

I had a much harder situation. Rolling 4d6 keep best 3 nine times for six stats and my highest roll was a 5. Had several threes.

There's a rule from 3rd edition that had fixed this kind of issue. Players are entitled to a stat reroll if either of the following conditions apply: the sum of all stat modifiers is +3 or less, AND/OR the highest stat is 13 or lower. It pretty much guarantees you end up with a stat array that's at least playable, even if not great. And you can use it with basically any rolling method too.


Ever since 5e came out, I've been really surprised that they got rid of the rule. I strongly recommend it for any group that won't allow players to finish character creation with garbage stats.

[Strigon]

I think he's just saying that it's possible that he might have genuinely rolled 18s.

It most certainly is possible. Just like it's possible that I could guess your email and password by bashing my hands randomly on the keyboard. That doesn't mean we should entertain the possibility.

[thedanster7000]

It most certainly is possible. Just like it's possible that I could guess your email and password by bashing my hands randomly on the keyboard. That doesn't mean we should entertain the possibility.
Precisely.

[Mastikator]

The odds are 100% if you pick wizard. 0% if you don't. :smallamused:

[Quertus]

On the other hand, low odds don't mean impossible. Try this little experiment at home. Flip a coin 20 times and write down the sequence of heads and tails results that you get. The odds of getting that exact sequence are about 1 in a billion, but you just did it. Every sequence has the same chance as all heads or all tails, but we only attach importance to the ones that look unique even though they're all unique.

And you did it on your first try!


I had a much harder situation. Rolling 4d6 keep best 3 nine times for six stats and my highest roll was a 5. Had several threes.

Congratulations, I've only known 2 people IRL with your luck (or lack thereof) with dice.

Anybody care to calculate the probability on this one?


It most certainly is possible. Just like it's possible that I could guess your email and password by bashing my hands randomly on the keyboard. That doesn't mean we should entertain the possibility.

... Stranger things have happened. So it's silly to simply discount the possibility. But, yes, it certainly is sufficiently improbable as to raise some red flags.

[Jay R]

I always roll my stats in front of the DM (or another witness he or she trusts) simply because someday I may hit the jackpot and get two or three 18s. The day that happens, I want people to know I'm not cheating.

[Slipperychicken]

I always roll my stats in front of the DM (or another witness he or she trusts) simply because someday I may hit the jackpot and get two or three 18s. The day that happens, I want people to know I'm not cheating.

It's just a best practice. Everyone should be doing it to help ensure the authenticity of rolled stats. The only thing I add to that is for the GM to record each player's stat rolls somewhere for later comparison.

[FreddyNoNose]

I always roll my stats in front of the DM (or another witness he or she trusts) simply because someday I may hit the jackpot and get two or three 18s. The day that happens, I want people to know I'm not cheating.

I do the same thing plus I require players to roll in front of me.

[FreddyNoNose]

There's a rule from 3rd edition that had fixed this kind of issue. Players are entitled to a stat reroll if either of the following conditions apply: the sum of all stat modifiers is +3 or less, AND/OR the highest stat is 13 or lower. It pretty much guarantees you end up with a stat array that's at least playable, even if not great. And you can use it with basically any rolling method too.


Ever since 5e came out, I've been really surprised that they got rid of the rule. I strongly recommend it for any group that won't allow players to finish character creation with garbage stats.

it doesn't need a fix because it wasn't a problem. DM simply said the character died in childhood and I rolled another character. Don't need rules for every little bit. Not a fan of the newer dnd systems (AKA anything newer than 2nd edtion).

[veti]

I had a much harder situation. Rolling 4d6 keep best 3 nine times for six stats and my highest roll was a 5. Had several threes.

In 1st edition, that character would be legally unplayable. All classes require an official minimum of 6 in (at least) all but one stat - there is no class you could play with two or more stats at 5 or less.

[Kelb_Panthera]

The odds are 100% if you pick wizard. 0% if you don't. :smallamused:

LOL no. There's no such thing as a perfect D&D character. Closest you get without going full pun-pun is a spell to power erudite that's been pushed to the optimization limits and even he can fall to one of the other T1 classes under the right circumstances.

Not that this is at all relevant to the OP's question. (misleading thread title.)

[Solaris]

I always roll my stats in front of the DM (or another witness he or she trusts) simply because someday I may hit the jackpot and get two or three 18s. The day that happens, I want people to know I'm not cheating.

My brother did this when he rolled a character with straight 18s for stats.
He then re-rolled on account of wanting at least some challenge. His previous character 'only' had four 18s, a 16, and a 15, and it pretty much dominated fights.

[FreddyNoNose]

In 1st edition, that character would be legally unplayable. All classes require an official minimum of 6 in (at least) all but one stat - there is no class you could play with two or more stats at 5 or less.
I know. So?

[Tanarii]

I had a much harder situation. Rolling 4d6 keep best 3 nine times for six stats and my highest roll was a 5. Had several threes.
Congratulations, I've only known 2 people IRL with your luck (or lack thereof) with dice.

Anybody care to calculate the probability on this one?The mass probability function is n!/[k!(n-k!)] * p^k * (1-p)^(n-k), where n is number of trials (6), k is number of successes, and p is probability of success.

The chance p of getting a 6+ on a single 4d6b3 is 98.84%. That means your chance of getting at least 1 6 or higher in 6 rolls is: .9884 * (.0116)^6 = 2.5*10-10%, or about 1 in 400 Billion.

Edit: Oops. Missed it was 9 rolls. That's .9984*(.0116)^9, or 3.75*10-18, or 3 in 1 quintillion.

[Tanarii]

Conversely the odds of one 17 or lower on 4d6b3 = 98.38%.

So in 6 trials, your odds of getting at least one 17 is .9838 * (1-.9838)6, or ~1.8*10-11, or about 1 in 50 Billion.

[Quertus]

The mass probability function is n!/[k!(n-k!)] * p^k * (1-p)^(n-k), where n is number of trials (6), k is number of successes, and p is probability of success.

The chance p of getting a 6+ on a single 4d6b3 is 98.84%. That means your chance of getting at least 1 6 or higher in 6 rolls is: .9884 * (.0116)^6 = 2.5*10-10%, or about 1 in 400 Billion.

Thanks! One of my many flaws is that I don't believe in memorizing or even looking up formulas, but deriving them from scratch whenever I need them. This practice worked out much better for me when I was younger, and using math all the time... and when the internet wasn't a thing.

[Tanarii]

Thanks! One of my many flaws is that I don't believe in memorizing or even looking up formulas, but deriving them from scratch whenever I need them. This practice worked out much better for me when I was younger, and using math all the time... and when the internet wasn't a thing.
Yeah I had to look it up online and figure out the probability mass function a while back. It's a pretty good formula for determining the chance of, for example, rolling 3 or more 16s.

The chance of rolling dice I just pull from anydice using "output [highest 3 of 4d6]" in combination with at most and at least.

[Mechalich]

One interesting thing about these odds is that, if you take them at face value, it means the more people there are in a setting, the more likely you are to have characters with supremely ridiculous starting stats.

For example, the average D&D setting might have 1 billion people on the planet and significantly less on a given continent. The Forgotten Realms has maybe around 200 million in its core regions. A modern setting could have 6 or 7 billion. So something that's a one in a billion chance probably doesn't exist in the Forgotten Realms, but has almost certainly occurred a couple of times on a faux Earth modern setting. Going further, a galaxy spanning space opera can have an eye-popping number of beings. Star Wars has a figure of 100 quadrillion (10^17) inhabitants.

If you assume that extraordinary ability is correlated with extraordinary achievement, the likelihood of people with extreme stat arrays rising to the top increases considerably the more people you toss into the system (assuming they are able to move freely of course). So a hypothetical Emperor Palpatine or Boba Fett might actually have multiple natural 18s.

[Solaris]

Which is itself an argument for using stat generation more favorable than 4d6 drop low to generate PCs, considering the explicit assumption that they're exceptional within the setting.

[D+1]

That is correct. Or technically it's 0.000000000167, but potayto potahto.

Of course, due to how many characters must have been rolled up since the invention of D&D, it's probably happened at some point. That's the law of truly large numbers.Read this in a 4-page story once and it has stuck with me to this day:
"...When you back odds like that and win, the thing to do is to look for some factor that is cheating in your favor."

By the way, if your player is trying to cheat by claiming he rolled all 18s, he's kind of an idiot. At least fake something more probable.
So, yeah, it's possible. It's just so ridiculously IMprobable it DOES NOT warrant consideration as being true. Not for a moment.

[FreddyNoNose]

Which is itself an argument for using stat generation more favorable than 4d6 drop low to generate PCs, considering the explicit assumption that they're exceptional within the setting.

It depends on how you want to look at exceptional. regular human npcs running around are not created using 3d6. They also don't get PC hit dice even if you decide to give farmers levels, they aren't going to be using D10s. PCs are exceptional over the regular folk using the regular 3d6. What I think you are confusing is the players desire for stats bonuses and justifying it using "the players are exceptional" bit.

I understand many players want (AKA NEED) high stats and bonuses. I understand how "unfair" it is for them not to have them. This is especially true after decades of gaming reinforcing the min-max therefore I want max stats mindset.

[Belac93]

What are the odds of getting three 18s and three 17s on 4d6b3? Best I ever rolled.

[meschlum]

The mass probability function is n!/[k!(n-k!)] * p^k * (1-p)^(n-k), where n is number of trials (6), k is number of successes, and p is probability of success.

The chance p of getting a 6+ on a single 4d6b3 is 98.84%. That means your chance of getting at least 1 6 or higher in 6 rolls is: .9884 * (.0116)^6 = 2.5*10-10%, or about 1 in 400 Billion.

Edit: Oops. Missed it was 9 rolls. That's .9984*(.0116)^9, or 3.75*10-18, or 3 in 1 quintillion.

Well, no. Given the probability of success (rolling a 6 or more), then the probability of failure after N trials is (1 - p)^N. No extra multiplication by .9884 involved - you may be trying to get the odds of having exactly one success out of N trials? In which case you'd need to multiply by 6 (or 9), and have one less factor for (1-p).

Still tiny, but now accurately so!


And your odds with 17s are just wrong.

Assuming the odds of a 17 or less are p (98.38%), I'm guessing you're after the odds of getting at least one 18? Because 17 or more isn't well informed by the probability of 17 or less - you might roll a 17!

The probability of getting all 6 (or 9) rolls equal to 17 or less is simply 0.9838^6 (or 9), so 90.67% (86.33%), meaning you get at least one 18 with probability 9.33% (or 13.67%). I think you were trying to compute the odds of getting exactly five 18s, and rolling too many times.


In fact, let's look at rolls of 18 with 4b3: you need at least 3 sixes. So you have two cases:

- Exactly 3 sixes and one non six, with probability (1/6)^3 * (5/6) * 4
- Exactly 4 sixes, with probability (1/6)^4

For a total of 21 / 1296 = 1.62%. So the probability of 17 or less is 1 - 1.62% = 98.38% The system works!

Now let's look at rolls of 17 with 4b3: you need exactly 2 sixes and one (or more) five(s). So you have these cases:

- 2 sixes and 2 fives, with probability (1/6)^2 * (1/6)^2 * 4 * 3 / 2
- 2 sixes, exactly 1 five, and one non five or six, with probability (1/6)^2 * (1/6) * (4/6) * 4 * 3

For a total of 54 / 1296 = 4.167%. So the probability of a 17 or more on 4b3 is 75/1296 = 5.787%. Slightly better than a natural 20. And the probability of rolling 16 or less is simply 94.213% (1 - 5.787%).

Combinatorics isn't hard, per se. It's just easy to lose track of the operations you're performing.



Edit: and to answer the question of the odds for three 18s and three 17s: the probability is simple!

(21 / 1296)^3 * (54 / 1296)^3 * 6 * 5 * 4 / (3 * 2 * 1) = 6.16e-9, so one in 162 million or so. Outright possible on the scale of the Earth!

[Batou1976]

Let me take a crack at this calculation.

<snip>

Who says math isn't fun?

I do. :smallfrown:

[Tanarii]

Well, no. Given the probability of success (rolling a 6 or more), then the probability of failure after N trials is (1 - p)^N. No extra multiplication by .9884 involved - you may be trying to get the odds of having exactly one success out of N trials? In which case you'd need to multiply by 6 (or 9), and have one less factor for (1-p).

Still tiny, but now accurately so!Good catch. I'm working off of a looked up formula, not a deeply comprehended understanding of the looked up math. Even so it looked a little weird to me when I was running it.


And your odds with 17s are just wrong.

Assuming the odds of a 17 or less are p (98.38%), I'm guessing you're after the odds of getting at least one 18? Because 17 or more isn't well informed by the probability of 17 or less - you might roll a 17!No, I was calculating the odds of getting at least one 17 or lower, possibly more. Again though, I believe you're right and I instead calculated the odds of getting exactly one 17 or lower.

[Solaris]

It depends on how you want to look at exceptional. regular human npcs running around are not created using 3d6. They also don't get PC hit dice even if you decide to give farmers levels, they aren't going to be using D10s. PCs are exceptional over the regular folk using the regular 3d6. What I think you are confusing is the players desire for stats bonuses and justifying it using "the players are exceptional" bit.

I understand many players want (AKA NEED) high stats and bonuses. I understand how "unfair" it is for them not to have them. This is especially true after decades of gaming reinforcing the min-max therefore I want max stats mindset.

I confuse nothing, and your contempt for the players who want to play decent stats betrays an ignorance of 3.X.
Take a gander at the paladin or monk classes of 3.5E some time. Then look up 'linear fighters and quadratic wizards'. The most powerful spellcasters actually benefit less from high ability scores (outside of their casting stat, of course) than the lower-tier MAD classes do - and if you're min-maxing using anything but a primary spellcaster in those games, you're doing it wrong. It's not 'min-maxing' or 'power-gaming', it's a basic comprehension of the math involved with the game.

The games are built around the assumption that exceptional characters will have exceptional stats - that's why it recommends higher points for higher-power games. That's also why a number of classes simply do not work with mediocre ability scores.

[Efrate]

Could someone post or cite the official stat mod totals must be +3 or higher rule? I know when your highest stat is a 13 you get the reroll but do not remember seeing anything about total mods being +3 or higher. I have a friend in a game I'm playing whose high stat is a 15 and the the rest are +0. He agreed to play it, but he is hating his rolls.

[FabulousFizban]

All my characters are perfect :P

[Yukitsu]

As a side comment, you could roll all 18s and then badly flub the build in general and end up with something that isn't good at all.

The odds of a person rolling all 18s and then being capable of creating a perfect character with them may be infinitely small.