Critical Streaks (1's & 20's)

[Devigor]

Alright, so a few sessions ago, I rolled 11 natural 1's in a row.
Today, I rolled 48 natural 20's in a row... Out of somewhere between 50 to 60 d20 rolls.
I swapped out dice repeatedly, in both instances, and still had this continuous weirdness.

Anyone else have wacky luck, or an idea to fix this?

It threw the game completely. Our party is level 5. The end "boss fight" with two level 10 elite green dragons lasted three rounds. I was the only one hurt, and I still had more than a surge worth of HP left.

[Emperor Ing]

You got 48 out of >60 rolls on your d20 to land on the 20.

...and you want to fix this?

Teach me your ways, Master!

[golentan]

Alright, so a few sessions ago, I rolled 11 natural 1's in a row.
Today, I rolled 48 natural 20's in a row... Out of somewhere between 50 to 60 d20 rolls.
I swapped out dice repeatedly, in both instances, and still had this continuous weirdness.

Anyone else have wacky luck, or an idea to fix this?

It threw the game completely. Our party is level 5. The end "boss fight" with two level 10 elite green dragons lasted three rounds. I was the only one hurt, and I still had more than a surge worth of HP left.

How confident are you that nobody microwaved your dice?

[Flickerdart]

“A spectacular vindication of the principle that each individual coin spun individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does.”

[golentan]

“A spectacular vindication of the principle that each individual coin spun individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does.”

The chance of that many sequential 20s on truly random dice rolls is 1 / 2.8147498e+62
28147498000000000000000000000000000000000000000000
The percentage chance of it occurring is so small my calculator won't track it. If every one of the approximately 100 billion homo sapiens who ever lived had a d20 and rolled it every day of their lives, you would not expect that result to come up even once. Yes, there is a small chance of it occurring, but a more likely explanation is someone weighted the dice very effectively.

[AtlanteanTroll]

So is this actually statistically significant?

[factotum]

See, if I'd done something as unlikely as that, I'd be cursing my luck that it didn't happen when picking my lottery numbers for the week. :smallbiggrin:

[Flickerdart]

The chance of that many sequential 20s on truly random dice rolls is 1 / 2.8147498e+62
28147498000000000000000000000000000000000000000000
The percentage chance of it occurring is so small my calculator won't track it. If every one of the approximately 100 billion homo sapiens who ever lived had a d20 and rolled it every day of their lives, you would not expect that result to come up even once. Yes, there is a small chance of it occurring, but a more likely explanation is someone weighted the dice very effectively.
Your reference game is poor, sir.

[Razanir]

So is this actually statistically significant?

(This math is not going to take into account the fact that they were all in a row. I don't feel like remembering the streak test)

Let's see... 48/60 dice rolls being natural 20s. I feel like 60 could be enough for the normal approximation, but the explanation will be more satisfying to me if I don't use it.

X~Binom(60,1/20)

P(X=48) = 2.86e-51

The mode of the binomial is (n+1)p = 61/20 = 3.05. 48 >> 3, so I can guarantee you that getting MORE 20s will be even less likely. So:

P(X>=48) = P(X=48) + P(X=49) + ... + P(X=60) < P(X=48) + P(X=48) + ... + P(X=48) = 13*P(X=48)

I won't even bother with actual numbers at this point, and I'll just say that P(X>=48) is on the order of 10^-50. Which is very, very, VERY tiny. MUCH smaller than the normal 1% or 5% chance normally used as a cutoff.

So yes. It's statistically significant.

[golentan]

Your reference game is poor, sir.

Your reference was not particularly conducive to understanding the situation.

[sktarq]

Anyone else have wacky luck, or an idea to fix this?

One option is to just roll with it. These things happen.
Option two-call it running too long at somepoint...have another person roll the dice for someone a couple times. It'll be good for laughs at least.
Option three twist it. Super successes and super fails that are just odd and fun... up to the ST/GM/DM...I had d10 based system throw me up 7 ones out of nine dice on a invisibility power....so the character got everyone's attention. Lots of fun.

[Jay R]

“A spectacular vindication of the principle that each individual coin spun individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does.”

"I've never known anything like it."

[Feytalist]

So is this actually statistically significant?


So yes. It's statistically significant.

...no.

This is what we in the financial statistics business call "spurious bias" (a sub-section of "spurious selection").

See, we humans tend to attach a higher significance to certain groupings of things than other, statistically equivalent, groupings. The chance of getting forty-eight 20's in a row, is the same as getting forty-seven 20's in a row and one 19. And we wouldn't regard the latter as significant (well, we might regard it "less" significant somehow). In a game such as D&D, the result is obviously beneficial to you. But then again, probably so would a random jumble of 18s, 19s and 20s. And most likely we wouldn't even notice the latter.

The result is simply rare. But then, *any* specific order of dice rolls would be similarly rare.

(By the way, I liked that Razanir did a significance test. So the result actually *is* statistically significant, but probably not in the way we think. If we were hypothesising that the dice used was biased to land on the 20 somehow, the result would be significant in that sense. Not certainly not in a general sense.)

[Razanir]

(By the way, I liked that Razanir did a significance test. So the result actually *is* statistically significant, but probably not in the way we think. If we were hypothesising that the dice used was biased to land on the 20 somehow, the result would be significant in that sense. Not certainly not in a general sense.)

(I'm practically majoring in stats. Significance tests are second nature to me)

Right. Especially after your explanation of spurious bias, I know the test I ran is really to see if the probability of a 20 is actually 5%. I just didn't think to make that distinction at the end of my explanation.

So I suppose a more accurate way to word my conclusion would be:

The probability of getting so many 20s in only 60 rolls is so low that I am willing to say at α=.01 that your die is weighted toward 20s. (Although as a note to non-stats people, this is not the same as saying it IS weighted. There's always the chance I made a Type I error)

[Jay R]

To explain the esoteric point that they are discussing:

A "statistically significant" result is one that, at the agreed upon level of significance, is grounds for rejecting the null hypothesis and accepting the alternate hypothesis.

There is no such thing as a "statistically significant" result unless you are going through the process of a hypothesis test, have formulated null and alternate hypotheses, and chosen a level of significance.

Discussing whether a result is statistically significant in the absence of a hypothesis test is like discussing whether an action is unlawful when no laws have been passed, or asking how much you like the movie without picking a movie first.

[Razanir]

To explain the esoteric point that they are discussing:

A "statistically significant" result is one that, at the agreed upon level of significance, is grounds for rejecting the null hypothesis and accepting the alternate hypothesis.

There is no such thing as a "statistically significant" result unless you are going through the process of a hypothesis test, have formulated null and alternate hypotheses, and chosen a level of significance.

Discussing whether a result is statistically significant in the absence of a hypothesis test is like discussing whether an action is unlawful when no laws have been passed, or asking how much you like the movie without picking a movie first.

To elaborate for this case:

(Note: I'm in computer science, so != is "not equal")

We are interested in seeing if 20s really do come up 5% of the time. In theory, they do. So our null hypothesis is p=.05. Our alternative hypothesis, then, is p!=.05. Now, by a different rule in stats, the alternative hypothesis must be true, since the probability a continuous value (ANY number in a range) is exactly one value is 0. But for the purposes of our test, we ignore that.

Instead, what we do, is assume the null IS true. Then we can calculate how likely it is that we get at least extreme of a value as we did. And if that probability is low enough, we say we have statistically significant results. This doesn't mean the null hypothesis IS false. It only means we have reason to assume it isn't. This probability is typically .05, .01, or even .005. Although particle physics is known to use alphas as small as 10^-7 (5σ).

So in this case, that probability is about 10^-50. That is ludicrously small. That is millions of times smaller than even the already super low alpha level used in physics. In other words, we've definitely seen something significant.

Because of that, we can say we have reason to believe the die is unfairly weighted toward 20s. But there's always the chance you did just get insanely lucky.

And since you're probably wondering, we CAN, in fact, estimate the actual probability. A 95% confidence interval for p would be [.6988,0.9012]. The 95% does not, however, mean there's a 95% chance the actual probability is in the range. It means that 95% of the possible intervals calculated that way (among all the possible rolls of 60 dice) will contain the actual probability. Either way, though, we're definitely estimating that your die is a biiiit weighted.

[sktarq]

Either way, though, we're definitely estimating that your die is a biiiit weighted.
Dice.


Alright, so a few sessions ago, I rolled 11 natural 1's in a row.
Today, I rolled 48 natural 20's in a row... Out of somewhere between 50 to 60 d20 rolls.
I swapped out dice repeatedly, in both instances, and still had this continuous weirdness.

That it happened also with a streak of 1's is only interesting if A: they are the same dice as a die is unlikely to be weighted to both 1 and 20 as they are opposite sides of the icosahedron that is the d20. - It would involve a system of rings of more dense material around the extreme values which would be harder to create either on purpose or accidentally during the molding process. B:You (knowingly or not) or the environment have a way effecting the outcome of the dice outside the the dice themselves that is creating a bias. Field interactions interacting with a particular dice rolling box and magnetized dice would be the type of thing I'm talking about but also highly unlikely. That said we are dealing with a result that is on its face unlikely and so all options are going to be low probability events. A biased die is a low probability event.

[TheEmerged]

Any time this comes up, I have to mention my Risk(tm) experience. After a turn-in of cards, I was attacking Iceland (with 2 armies) from Greenland (with 43 attacking armies). There was only 1 other property on the board I didn't control (Great Britain if I remember), and the defending player was going to turn in cards next turn if I didn't win. Basically I win this turn or lose next turn\shortly thereafter.


You can roll 1 die for each attacking & defending army, up to 3 for the attacker and 2 for the defender. Highest result wins, tie goes to defender. If the defender rolls two dice, the highest attacker die is compared to the highest defender die, and then the second-highest attacker die is compared to the remaining defender die. The loser removes one army per pair of dice compared - so if the defender rolls 2 dice two armies are removed (it can split 1 each), and if the defender rolls 1 die 1 army is removed. Combat then proceeds to the next round unless the attacker bows out.


The first roll (3 dices versus 2) split, so it was now 42 armies attacking 1.

I proceeded to lose the next 42 rolls, and then the game within a couple of turns.

These were *not* loaded dice as they were the ones that came with the game and I was the owner of said game. Further they had behaved normally up to this point and continued behaving normally afterwards.

================================================== =

In terms of actual "crit streaks", I have had a couple of cases where I've rolled almost 50% critical hits when DMing.

[Jay R]

To elaborate for this case:

(Note: I'm in computer science, so != is "not equal")

We are interested in seeing if 20s really do come up 5% of the time. In theory, they do. So our null hypothesis is p=.05. Our alternative hypothesis, then, is p!=.05. Now, by a different rule in stats, the alternative hypothesis must be true, since the probability a continuous value (ANY number in a range) is exactly one value is 0. But for the purposes of our test, we ignore that.

Instead, what we do, is assume the null IS true. Then we can calculate how likely it is that we get at least extreme of a value as we did. And if that probability is low enough, we say we have statistically significant results. This doesn't mean the null hypothesis IS false. It only means we have reason to assume it isn't. This probability is typically .05, .01, or even .005. Although particle physics is known to use alphas as small as 10^-7 (5σ).

This is a good description of what happens in an experiment. It does not, however, describe what happened in this case, which was not an experiment. The dice were used for a long time, and then when a short sequence of 60 rolls seemed unusual, that sequence was tested alone, without the previous rolls included.


So in this case, that probability is about 10^-50.

No, it isn't. That describes a short sequence that was noticed only because it was unusual for the dice in question.

To run an experiment, you decide to run the experiment, and then take measurements, and use all of them (except known errors, such as when your two-year-old rolls it over to 7 each time).

You would never have calculated a p-value if this had been a normal sequence of rolls. So you are using a biased sample - biased by the action of choosing only an unusual run.

All of the calculations, which would be correct if you had made them on a randomly chosen sequence of rolls, are meaningless when applied to data chosen because it was unusual.


Because of that, we can say we have reason to believe the die is unfairly weighted toward 20s. But there's always the chance you did just get insanely lucky.

You can conclude that this was an extremely unlikely sequence of rolls. But without a bias-free sample, you cannot conclude anything about the dice.

[Razanir]

I used the data given. Would they hold up as an actual experiment? Of course not. Does it still serve to illustrate what statistical testing is like? Of course. But you're certainly right that testing only after we saw an incredibly unlikely event reeks of bias, compared to deciding the test first and rolling second.

[The_Ditto]

Well, first game in 5E on the weekend, I built an archer:

First roll: Initiative: rolled a 2.
First shot in combat: Rolled a 2. O.o
(short combat, over ..)

Second fight:
initiative: Rolled a 1. :smallfurious:

Not the greatest start, but things starting picking up after that. *shrug*

[TheThan]

I once lost an entire squad of 10 grey knight terminators to a blast template.
If you don't know Terminators have a 2+ armor save. That’s right, I rolled ten 1s in a single roll (ten dice all at once).

[Devigor]

This is awesome, how responsive everyone has been. Thanks.

I played another session with a Wizard's test group, different dice. 5th edition.
22 out of 27 rolls were natural 20's. The rest were all above 15.

I made the Wizards DM mad. We finished that session in half the time he prepared for. Then he sent the elder green dragon. I choppped its head off.

Fighters are much more fun in 5th than ever before. The mooks had to crit to hit me. None did.

[sktarq]

how long has this particular dice running to extremes been with you? As much as I consider it statistically unlikely I have notices certain players do seem to up with extremes more often than others for no reason I can adequately explain. Two notably. I had an acquaintance (friend of our DM) who in 2e never rarely had a character without an 18, almost never had any scores below 12 and often had a average in the 15-16 range. He would use other people's dice and still come up with a similar result. Over time he started all but rejecting normal scored characters and driving them into very high risk situations in order to move to better rolled backup characters which has some influence on how much I remember him playing characters with striking luck. The other I wouldn't say her average roll is any better or worse than normal but she does roll the extremes when it is interesting. Particularly when I (her ST) need something random - (are there any cars in the parking lot at the moment? I hadn't thought about that- their could be, or not-lets roll to see how busy it is-rolls a ten (tens are considered a success and allow for a reroll to obtain additional success -for oddball rolls it takes it out of expected territory and the rerolls say how far), rerolls another ten, rerolls another ten-then get a 8 ....and now the parking lot is packed and more cars are lined up down the street - and I need a reason....). Having her around makes the game interesting and she regularly derails lines of group directed "plot" by total accident.

[Zrak]

Your reference was not particularly conducive to understanding the situation.

Well, there are other possibilities he failed to mention.

One: he was willing it. Inside where nothing shows, he was the essence of a man rolling twenty-headed dice, and betting against himself in private atonement for an unremembered past.

Two: time has stopped dead, and the single experience of one die being rolled once has been repeated forty-seven times ... On the whole, doubtful.

[Ifni]

It wasn't a d20 system, but I was pretty happy when a very powerful enemy warrior tried to behead my geeky little follower of a goddess of fate*, rolled 17 dice (10-sided dice, a roll of 7 or higher is a success, 6 or lower is a fail) and failed on ALL of them :smallwink:

And it was an online roller, so no weighted dice.

*Who was trying to do the in-setting equivalent of defusing a bomb at the time, and not defending herself at all.

[golentan]

It wasn't a d20 system, but I was pretty happy when a very powerful enemy warrior tried to behead my geeky little follower of a goddess of fate*, rolled 17 dice (10-sided dice, a roll of 7 or higher is a success, 6 or lower is a fail) and failed on ALL of them :smallwink:

And it was an online roller, so no weighted dice.

*Who was trying to do the in-setting equivalent of defusing a bomb at the time, and not defending herself at all.

Fate conspires to protect sidereals, obviously. Did he botch?

[Ifni]

Fate conspires to protect sidereals, obviously. Did he botch?

Yes, yes he did. Poor Dawn caste.

(And yeah, for Exalted players, my PC was a Bronze Chosen of Secrets.)

[golentan]

Yes, yes he did. Poor Dawn caste.

(And yeah, for Exalted players, my PC was a Bronze Chosen of Secrets.)

Well, your situation would happen about 3 times in 20,000, so it's not as ludicrous, but I imagine it felt pretty good!

[Ifni]

Well, your situation would happen about 3 times in 20,000, so it's not as ludicrous, but I imagine it felt pretty good!

It was especially entertaining as she'd just made a successful prayer roll to Jupiter. (And when he missed and dropped his sword, she said something along the lines of "There's no point opposing fate". Did I mention this was during the Usurpation?) As you say, it's not that ludicrous, but the situation was very apropos.

[Devigor]

how long has this particular dice running to extremes been with you? As much as I consider it statistically unlikely I have notices certain players do seem to up with extremes more often than others for no reason I can adequately explain. Two notably. I had an acquaintance (friend of our DM) who in 2e never rarely had a character without an 18, almost never had any scores below 12 and often had a average in the 15-16 range. He would use other people's dice and still come up with a similar result.

Yup. I have been rolling like this for months. With many many sets of dice.
I have rolled up 7 characters (at different times) that have had straight 18's in the last few months, right in front of my DM's. The lowest set of abilities I have gotten for a year or so is Str 11 Con 14 Dex 14 Int 17 Wis 9 Cha 5.

[sktarq]

I have rolled up 7 characters (at different times) that have had straight 18's right in front of my DM's.
...It doesn't make a crit any more critical, or turn a roll into a crit, etc.
Take video. Send to DARPA. You'll either make millions or die in horrible experiments.
Or go to Vegas. . . play Craps

[Devigor]

Take video. Send to DARPA. You'll either make millions or die in horrible experiments.
Or go to Vegas. . . play Craps

Good idea. I think I'll do that. The first one, not gambling. The first time I gamble will be when I lose my luck.