Koumei's Dice Odds / Shadow's Trinity Probability Tested

[Zossie]

The following contains an excerpt from a much larger analysis of Koumei’s Omamori ability which I am currently writing up. However, after recently watching a Sabuuchi video in which he discussed the Triple Six bonus, also known as Shadow’s Trinity, and remarked with a post from DE Pablo that the probability is not the theoretical rate of 3.55%, assuming fair dice, I felt it pressing to share my findings. I’m not big into social media, so if a statement has been made by Pablo or any other devs regarding the probabilities of Koumei’s dice, feel free to leave that information in a reply.

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

Part 1 - “Fair Dice”

When Pablo described Koumei’s dice rolling bonus to her abilities, my inner mathematician immediately went to calculate the odds of receiving said bonuses. Anyone who has played Yahtzee already knows the rarity with which you can roll three-of-a-kind, and even rarer still to roll a three-of-a-kind specifically in Sixes. Utilizing https://anydice.com/program/ac03 we can calculate the likelihood of rolling at least 3 Sixes via:

output [count 6 in 5d6]

The resulting probability is 3.55%. Essentially, if using “Fair Dice,” one would expect to receive an ability enhancement in about 1 in every 33 ability uses. “Fair Dice” refers to each possible outcome of each die roll to have equal likelihood, or a uniform outcome probability.

However, over the course of testing with Koumei, I found that the Triple-Six rolls were occurring at a rate significantly higher than the expected 3.55%, even if only anecdotally. At current, there is no documentation which expresses the outcomes of Koumei’s dice rolls, or even if she is using “Fair Dice.” I hypothesized: Koumei is using Loaded Dice, enhancing the likelihood of rolling a Triple-Six Outcome. It was time to put my hypothesis to the test. I created the following spreadsheet:

The first sheet tab, titled “Fair Dice Rolls,” emulates rolling a set of 5 “Fair Dice” 500 times. Across these rolls, the results are recorded in order to determine which rolls would trigger the Triple-Six bonus. After collecting the data, we see that 19 of the 500 rolls included at a minimum 3 Sixes, which comes out to 3.8% of the 500 rolls. This experimental 3.8% closely aligns with the calculated theoretical 3.55%, within acceptable statistical margin. Further, we can see the distribution of rolls in the histogram appears very close to a uniform distribution. The calculated roll percentages also indicate that each outcome has a likelihood of ~16%, or 1 in 6, as would be expected of a fair six-sided die. 

The second sheet tab, titled “In-Game Rolls,” records the outcomes of 500 sequential Omamori rolls in the Simulacrum over the course of approximately one and a half hours. Here is an image of only a portion of the rolls which triggered Shadow's Trinity in my time in the Simulacrum.

From these results it was found that the Triple-Six outcome occurred an astonishing 59 times, over three times the expected rate of “Fair Dice.” This equates to 11.8% of the 500 ability uses. Furthermore, both the single die and resultantly the Five Dice roll averages were greater than those of the “Fair Dice,” which is of consequence for abilities that care about the dice roll totals, such as Kumihimo’s Strength Scaling and Bunraku’s number of Status Procs. The histogram of these results shows that Koumei’s dice do not have a uniform distribution, and is instead weighted toward rolling Sixes. 

In Conclusion: It appears that Koumei’s dice rolling bonus on her abilities is weighted more in favor of rolling Sixes to increase the rate of receiving the Triple-Six bonus, far greater than statistically expected. Though untested by myself, I have no reason to believe that Koumei’s other abilities do not receive the same weighted benefit. Such a test could easily be performed using the quickly recastable Kumihimo, or even a replication of my test methodology for those who wish to verify the results through peer review. As a result of these findings, I would like to ask a dev to shed more light on the results by stating definitively if this was the intention and what the odds are. DE already releases the drop tables for items and mission rewards as well as the percentages for Relic rewards, so shedding some light on this aspect of RNG would be much appreciated and documented on the wiki.

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

Much time and three hotfixes have passed since I originally ran this test. The tests were conducted on game version 37.0.3. My Koumei was only modded with Streamline and Fleeting Expertise in an effort to reduce Energy Costs to expedite the length of the tests. I’m stating this on the off chance that any mods, such as those affecting Ability Strength, have an effect on the dice roll outcomes (though to my knowledge nothing of the sort has been officially stated).

 

The TL;DR of all this is, Koumei’s odds of rolling a Six on her dice, and consequently or receiving the Shadow’s Trinity bonus for rolling 3 or more Sixes, are greater than their respective theoretical probabilities of 16.67% and 3.55%, respectively. The tests conducted were able to generate experimental results with “fair dice” which fell within a 95% confidence interval of the theoretical values. 

From the in-game dice roll data collected, the following 95% confidence intervals were calculated:

   Odds of a single Koumei die rolling a 6:      23.76% +/- 1.67%

   Odds of rolling at least 3 Sixes in 5 dice rolled (Triggering Shadow’s Trinity on an ability cast):     11.80% +/- 2.83%

To clean all this up and put a bow on it: Any single die Koumei rolls has a ~24% chance of rolling a Six. As a result, the odds of triggering Shadow’s Trinity on an ability cast is ~12%.

At the time of writing, the Warframe Wiki for Koumei’s abilities posits a 3.55% chance of triggering Shadow’s Trinity. Between the experimental results and DE Pablo’s social media post explicitly stating that 3.55% is not the odds, I would encourage a contributor/editor of the wiki to include the following results citing this post as well as Pablo’s tweet, at least until such a time that a developer explicitly releases the actual odds. If the validity of the tests are called into question, or if anyone wishes to peer review the results by conducting the test for themselves, feel free to contact me for help regarding the experiment’s procedure, though it is fairly straightforward. I managed to screenshot 50+ of the dice rolls resulting in Shadow’s Trinity triggering and would have posted them all in the attached screenshot, but my photo editing software kept crashing after the 26 rolls shown. 

 

Anyway, thanks for reading.

[_Anise_]

I guess you posted this over on reddit too? (its the same link)

I already seen people do that math that Koumei isn't close to voruna or citrine when it comes to throwing around status effects I'm curious what you think about her being based around so much rng? I think it seems to make her inconsistent and imo you need some kind of stable foundations to balance something like this properly?

 

[Zossie]

1 hour ago, _Anise_ said:

I guess you posted this over on reddit too? (its the same link)

Wasn't me. As I said, I'm not big into social media, as you can probably tell from my low forum post count. But whatever gets the word out is fine, so long as no one is falsely trying to claim credit. 

As for the larger scope of her kit being based on RNG only slightly better than the odds of getting a Rare drop when running a solo Radiant Relic, there is a lot more to be said. I expect I'll touch upon it quite a bit in my larger analysis post (5 pages and still growing). I understand the intention of DE making Koumei accessible to new players, but creating her thematically based on luck really curbs her mod build-craft. Take for example her 2, Omikuji. There was a time when Garuda was the only Warframe on the wiki to have an ability which doesn't scale with Ability Strength, Range, or Duration. Fortunately for Garuda, it at least scales with Efficiency. But the award for non-modable abilities now rests firmly with Koumei. And on a practical level, this point of view even extends to her 3, Omamori. The wiki presents that Omamori only scales with Ability Strength, but is far more impacted by both her defensive mods (healthpool, shields, damage mitigation) and the enemy's damage. There is a very narrow window of content for which the health conversion is meaningful, and made even more narrow by the 50% trigger rate of Omamori. Below this level, incoming damage is too low to be considered a threat; and beyond that narrow scope, enemy damage will scale high enough to render Omamori's defenses irrelevant. You are either converting the damage in full, regardless of your Ability Strength scaling, or getting one shot if you've failed to keep up a meta damage mitigation like shield gating, overshield, or an allies defensive buff (Mesmer skin for example).

A new player after acquiring Koumei could easily see their way completely through the star chart with her. But her kit has so much parasitic design. All her abilities are completely reliant on her passive, not her build or her mods. Other Warframes have build diversity as they chase after specific stat breakpoints; to enable full armor strip, reach Damage Reduction caps, Nuke the precise area of Hydron, or maintain 100% buff uptime. It is very telling when Koumei's Abilities wiki page features more in the "misc" section than her"Strength/Range/Duration" sections.

 

I would have to agree on a lot of the community and even the creators' when they call her Cope-mei and Koumid.

[sXeth]

7 hours ago, Zossie said:

The following contains an excerpt from a much larger analysis of Koumei’s Omamori ability which I am currently writing up. However, after recently watching a Sabuuchi video in which he discussed the Triple Six bonus, also known as Shadow’s Trinity, and remarked with a post from DE Pablo that the probability is not the theoretical rate of 3.55%, assuming fair dice, I felt it pressing to share my findings. I’m not big into social media, so if a statement has been made by Pablo or any other devs regarding the probabilities of Koumei’s dice, feel free to leave that information in a reply.

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

Part 1 - “Fair Dice”

When Pablo described Koumei’s dice rolling bonus to her abilities, my inner mathematician immediately went to calculate the odds of receiving said bonuses. Anyone who has played Yahtzee already knows the rarity with which you can roll three-of-a-kind, and even rarer still to roll a three-of-a-kind specifically in Sixes. Utilizing https://anydice.com/program/ac03 we can calculate the likelihood of rolling at least 3 Sixes via:

output [count 6 in 5d6]

The resulting probability is 3.55%. Essentially, if using “Fair Dice,” one would expect to receive an ability enhancement in about 1 in every 33 ability uses. “Fair Dice” refers to each possible outcome of each die roll to have equal likelihood, or a uniform outcome probability.

However, over the course of testing with Koumei, I found that the Triple-Six rolls were occurring at a rate significantly higher than the expected 3.55%, even if only anecdotally. At current, there is no documentation which expresses the outcomes of Koumei’s dice rolls, or even if she is using “Fair Dice.” I hypothesized: Koumei is using Loaded Dice, enhancing the likelihood of rolling a Triple-Six Outcome. It was time to put my hypothesis to the test. I created the following spreadsheet:

The first sheet tab, titled “Fair Dice Rolls,” emulates rolling a set of 5 “Fair Dice” 500 times. Across these rolls, the results are recorded in order to determine which rolls would trigger the Triple-Six bonus. After collecting the data, we see that 19 of the 500 rolls included at a minimum 3 Sixes, which comes out to 3.8% of the 500 rolls. This experimental 3.8% closely aligns with the calculated theoretical 3.55%, within acceptable statistical margin. Further, we can see the distribution of rolls in the histogram appears very close to a uniform distribution. The calculated roll percentages also indicate that each outcome has a likelihood of ~16%, or 1 in 6, as would be expected of a fair six-sided die. 

The second sheet tab, titled “In-Game Rolls,” records the outcomes of 500 sequential Omamori rolls in the Simulacrum over the course of approximately one and a half hours. Here is an image of only a portion of the rolls which triggered Shadow's Trinity in my time in the Simulacrum.

From these results it was found that the Triple-Six outcome occurred an astonishing 59 times, over three times the expected rate of “Fair Dice.” This equates to 11.8% of the 500 ability uses. Furthermore, both the single die and resultantly the Five Dice roll averages were greater than those of the “Fair Dice,” which is of consequence for abilities that care about the dice roll totals, such as Kumihimo’s Strength Scaling and Bunraku’s number of Status Procs. The histogram of these results shows that Koumei’s dice do not have a uniform distribution, and is instead weighted toward rolling Sixes. 

In Conclusion: It appears that Koumei’s dice rolling bonus on her abilities is weighted more in favor of rolling Sixes to increase the rate of receiving the Triple-Six bonus, far greater than statistically expected. Though untested by myself, I have no reason to believe that Koumei’s other abilities do not receive the same weighted benefit. Such a test could easily be performed using the quickly recastable Kumihimo, or even a replication of my test methodology for those who wish to verify the results through peer review. As a result of these findings, I would like to ask a dev to shed more light on the results by stating definitively if this was the intention and what the odds are. DE already releases the drop tables for items and mission rewards as well as the percentages for Relic rewards, so shedding some light on this aspect of RNG would be much appreciated and documented on the wiki.

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

Much time and three hotfixes have passed since I originally ran this test. The tests were conducted on game version 37.0.3. My Koumei was only modded with Streamline and Fleeting Expertise in an effort to reduce Energy Costs to expedite the length of the tests. I’m stating this on the off chance that any mods, such as those affecting Ability Strength, have an effect on the dice roll outcomes (though to my knowledge nothing of the sort has been officially stated).

 

The TL;DR of all this is, Koumei’s odds of rolling a Six on her dice, and consequently or receiving the Shadow’s Trinity bonus for rolling 3 or more Sixes, are greater than their respective theoretical probabilities of 16.67% and 3.55%, respectively. The tests conducted were able to generate experimental results with “fair dice” which fell within a 95% confidence interval of the theoretical values. 

From the in-game dice roll data collected, the following 95% confidence intervals were calculated:

   Odds of a single Koumei die rolling a 6:      23.76% +/- 1.67%

   Odds of rolling at least 3 Sixes in 5 dice rolled (Triggering Shadow’s Trinity on an ability cast):     11.80% +/- 2.83%

To clean all this up and put a bow on it: Any single die Koumei rolls has a ~24% chance of rolling a Six. As a result, the odds of triggering Shadow’s Trinity on an ability cast is ~12%.

At the time of writing, the Warframe Wiki for Koumei’s abilities posits a 3.55% chance of triggering Shadow’s Trinity. Between the experimental results and DE Pablo’s social media post explicitly stating that 3.55% is not the odds, I would encourage a contributor/editor of the wiki to include the following results citing this post as well as Pablo’s tweet, at least until such a time that a developer explicitly releases the actual odds. If the validity of the tests are called into question, or if anyone wishes to peer review the results by conducting the test for themselves, feel free to contact me for help regarding the experiment’s procedure, though it is fairly straightforward. I managed to screenshot 50+ of the dice rolls resulting in Shadow’s Trinity triggering and would have posted them all in the attached screenshot, but my photo editing software kept crashing after the 26 rolls shown. 

 

Anyway, thanks for reading.

Its on Pablos twitter somewhere but yeah they’re fudged/loaded (intentionally) and 11.something for the triple sixes.

[Tiltskillet]

8 hours ago, Zossie said:

Anyway, thanks for reading

Good stuff.  You'd probably be interested in similar work done here: 

 

[_Anise_]

8 hours ago, Zossie said:

Wasn't me

oh my bad the link was similar not the same, its on the post that Tiltskillet linked

8 hours ago, Zossie said:

I expect I'll touch upon it quite a bit in my larger analysis post

I look forward to reading it! I think consistency is important essential aspect of design, it helps create a more engaging experience. while randomness can add excitement and variety, I think it's most effective when it contributes to emergent gameplay, where players can creatively interact with the game and its mechanics.

not the way Koumie does random, like the immortality on her 3, I can't rely on it, I can't plan for it so I don't, then when it finally happens it feels meaningless.

I recall one of the first times I modded a weapon when I had just joined warframe and showed a friend my build, asking if it was ok and I mentioned I didn't need any more of a particular stat (I think it was for multishot or status) because I already had 100%. I was surprised when he told me that you can exceed 100% with most stats in Warframe. I was so used to games having diminishing returns on stats that made it nearly impossible to achieve 100% caps, it was nice to see warframe didn't follow the same randomness and instead just added more tiers.

8 hours ago, Zossie said:

All her abilities are completely reliant on her passive, not her build or her mods.

makes me think a little of nezha, while he does scale with mods I completed the starchart using jank whatever build on him because his halo does almost all of the heavy lifting on keeping him alive, like I think it might even be possible to take him on steelpath without any survivability mods at all the only difference is halo will be removed more often but rather than being a random thing is more around him having a solid base kit that only gets better with mods.

8 hours ago, Zossie said:

would have to agree on a lot of the community and even the creators' when they call her Cope-mei and Koumid.

I did hear Cope-mei but Koumid is a new one! 🤣 I hope they fix her but the hotfix makes me think they are just going todo what they did to yareli and keep buffing her terrible design but unlike yareli I think people will play copemei as a weapons platform regardless.

Edited Saturday at 02:00 PM by _Anise_

[Zossie]

14 hours ago, Tiltskillet said:

Good stuff.  You'd probably be interested in similar work done here: 

Thanks for that link, I was searching reddit for it but couldn't find it. Dang, now I feel like slowpoke. Guess it's true what they say: on the internet, if you aren't first, you're last. VeeArr's test methodology, results, even data collection and spreadsheet formatting are so similar to mine, someone might expect foul play or plagiarism. But I'd chalk it up to "great minds think alike." If anything, it's great to see that the data enforces the conclusion. One point VeeArr raised is that by ignoring rolls which trigger Shadow's Trinity, the distribution of dice rolls is inline with a uniform distribution. I'll have to check if my data supports this assertion. And If it does, it may be the case that dice rolls may be predetermined to trigger Shadow's Trinity, irrespective of individual dice rolls, rather than the other way around. 

8 hours ago, _Anise_ said:

not the way Koumie does random, like the immortality on her 3, I can't rely on it, I can't plan for it so I don't, then when it finally happens it feels meaningless.

You absolutely get it. My Omamori write up is beginning to look more and more like a rant, and "feels meaningless" is often how I describe Omamori. There are instances in which you can game the recast mechanic to force the recast reset. If you are fortunate enough to be on a tileset with either Corpus Nullifiers or a kill volume that you can jump into (like a bottomless pit), you can clear the buff and simply recast Omamori until you receive Shadow's Trinity. The same could (and still can?) be done with Harrow's Covenant, utilizing the Invulnerability period, then clearing the buff to enable a recast, rather than waiting out the Retaliation period which is twice as long, during which Covenant cannot be recast.

You hit the nail on the head, "reliability" is important, and thanks to its binary nature which is entirely uninfluenced by mods, Omamori is anything but.

 

Edit: Just wanted to add my findings of testing VeeArr's assertion of dice rolls ignoring Shadow's Trinity results falling into a uniform distribution. I should correct myself here, as VeeArr states that their result "distribution is much more uniform," not explicitly concluding that the distribution is in fact that of fair dice. After striking the 59 Shadow's Trinity results from my data, rolls of Six still retained a statistically higher rate of ~20%. This result matches VeeArr's results as well, maintaining statistically high percentages across all test batches. As they posit, it is difficult to discern the exact method which has been implemented to raise the odds of triggering Shadow's Trinity when compared to fair dice. If the function is coded in a way that flatly triggers Shadow's passive ~12% of the time, one would expect the arbitrary "spare dice" of those rolls to fall into a uniform distribution. But based on the data, this doesn't seem to be the case. In fact, an alarmingly high amount of Fives was rolled in these cases, which lends credence to Fives rolling at a higher than normal rate for both my and all of VeeArr's test batches.

TL;DR - The most likely explanation is probably the simples. I conclude that the roll percentages are skewed more in favor of higher rolls (Sixes and Fives at ) as opposed to low rolls (Ones having the lowest percentage of ~13%).

Edited Saturday at 10:37 PM by Zossie

[0_The_F00l]

Great , so not only is koumei have poor modding potential.

She is a cheating gambler on top of that :D

Shame , shame.

 

Haha , jokes aside I am both disappointed and a little relieved.

Disappointed that the ability is rigged ,

Relieved cause the game has always been rigged , but for one of the few times it is rigged in favour of the player.

Now it's not even correct to call her a gambling frame , it's more accurate to call her swindler frame now.