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Summon Probabilities

This is a guide and calculator for the probabilities of hero summons in the noble tavern.

The data & concept for this page was contributed by Salkasm (Discord: Salkasm#4688) and kindly shared for the purpose of this page.

Probability Calculator

This is a probability calculator for summoning heroes in the noble tavern. It’s based on the EXACT amount of heroes you could obtain in 1, 5 & 10 attempts.

Due to rounding, probabilities below 0.05% are defined as 0%.

You have a 30.12% chance to obtain 2 Common hero(s) from 5 summon(s).

Formula

P is the probability of obtaining the exact success, n is the amount of tries (like 2700 Crystals is a pull of 10 heroes), k is the amount of successes (like, pulling 2 elite heroes in those 10 pulls) and x is the probability of the individual card (Common is 51.69%, Rare is 43.70% and Elite is 4.61%, as specified in-game).

When Can You Expect 1 Elite Hero?

On average, you can expect 1 Elite hero every 21.69 pulls (22 rounded upwards).

Thanks to BlackBacon, Kuroi & Xanzyl for helping correct this!

Pulling At Least 1 Hero

This is the probability to pull AT LEAST 1 hero of each rarity.

Hero PullsCommonRareElite
At least 1 in 597.37%94.34%21.03%
At least 1 in 1099.94%99.69%37.63%
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Beliar
Beliar
4 months ago

The ingame % in tavern is just a flat number provided (which is used conviniently to justify to people complaining about terrible drop rates). In reality its not so simple, and without a real formula server-side that happens in the background whenever the pull is issued we cannot calculate much useful data. What is 100% certain is that whenever a user pulls 1 or 10 summon, multiple checks on the server are made for each specific user. Example checks may be: – User.dateJoined (new players get a better chance, to lure them in game) – Tavern.wishlist (add small +% to… Read more »

Gallas
Gallas
5 months ago

I did a 10 pull twice today got 1 rare and 9 commons first pull 2nd I got 3 rare 7 common, then for whatever reason I pulled one at a time with 300 crystals 3 times and got 3 elite, one I had and two knew

Ice
Ice
Reply to  Gallas
2 months ago

You can get more than 7 commons its impossible,confirmed by lilith so stop lying

scooter
scooter
5 months ago

seems like everyones math is different. will we ever know the true odds? probably not. deal with it or don’t

Anh
Anh
5 months ago

I tried the formula but it only work within 10 tries. If I input 30 tries. The probabilities for 1 Elite card is 35.189% which is not correct.

Anh
Anh
Reply to  Anh
5 months ago

I found that the formula is for getting EXACTLY amount of cards. Is there any formula to get AT LEAST k cards?

Matt
Matt
Reply to  Anh
5 months ago

P(At least K heroes) = 1 – P(less than K heroes) = 1 – P(K=0) – P(K=1) – … – P(K=k-1)

Rage engine
Rage engine
5 months ago

I pulled 4 elites out of one ten pull this week

enteredname
enteredname
6 months ago

this is not very good RNG please up the number of elite you can get out of any rolls it should be at least above 40% t be plausible

tomoo
tomoo
7 months ago

i got 3 elites in one pool and usually get an elite when i do a 10 pull…

sora
sora
11 months ago

3 x 2700 = 1 hero elite ???

KevlarGolem
KevlarGolem
Reply to  sora
11 months ago

Yes if you do 1 more 10x pull and get an elite you will be slightly better than avg. on your 4 tries. And of course later if you pull two elites in a row or two in the same 10x you will immediately forget about it think that “should” happen when actually you were way above average.

Skylara
Skylara
11 months ago

I kept track on three accounts over 14 days. This was also exactly 100 draws, though some had more scrolls and other more companion draws.

Acc1 100 draws 6 elites 44 blues 50 greens
Acc2 100 draws 5 elites 46 blues 49 greens
Acc3 100 draws 5 elites 42 blues 53 greens

Each account got an elite from companion points, tho acc1 had only 20 draws while the others got 30.

2 accounts also got double elites in a 10x draw

Skylara
Skylara
Reply to  Skylara
11 months ago

Also all elites except one was from the wishlist. And another one because that was from companion points, and wishlist doesnt work for those. Ialso have only 2 graveborn in wishlist and i get one of those too.

Sandhi
Sandhi
11 months ago

In practice, it is actually very easy to determine. You can expect at least 1 elite in a x30 summons of the same type of scroll (barring companion summon).. So either a x30 faction scroll or x30 summon scroll (by scroll/dias), you cant mix both the scroll (eg. using a x20 ordinary summon scroll and then x10 faction scroll wont yield a guaranteed 1 elite, you will still have to draw x10 more on the ordinary summons and x20 faction summons). If you’re lucky you CAN get an elite on 1st or 2nd attempt, but the cutoff for a guaranteed… Read more »

VigasVelho
VigasVelho
Reply to  Sandhi
11 months ago

That is not how it works, mate.

Anthragg
Anthragg
Reply to  VigasVelho
9 months ago

He’s right, 3rd multi draws = 1 elite gueranteed. Even if it’s not written IG.

Jaycuttler
Jaycuttler
Reply to  Sandhi
11 months ago

Dude you slept deeply during math class…

Wattstrom
Wattstrom
11 months ago

A guild mate of mine emailed Lilith and according to the reply, every 3rd 10x pull from each ‘pool’ you are supposed to be guaranteed at least 1 Epic hero. A few of us in guild have figured out there’s apparently 3 different pools: Diamonds, Scrolls & Faction Scrolls. The pulls from companion points are completely separate from this system and doesn’t matter if you do single pulls or 10x pulls your chances are exactly the same so spend away. Hope this helps.

BlackBacon
BlackBacon
Reply to  Wattstrom
11 months ago

Wow, if this is true I’ve been using daily/weekly reward scrolls wrong this whole time by summoning individually. It certainly changes your odds.

Elites in 30 Pulls (Only using 10x)
Elites Odds
0(1) 24.27%
1 35.19%
2 24.66%
3 11.12%
4 3.63%
5 0.91%

If you’re only doing 10 pulls you should actually average 5.37% Elites which is 16.49% more elites than the 4.61% single pull rate. 1-(5.37/4.61)

Wattstrom
Wattstrom
Reply to  BlackBacon
11 months ago

Right?! It seems most of us have!

At first, we thought it was every 3 pulls of 10x no matter the source, but another person in guild did 3 and didn’t get an epic. When we got to talking about it, he used 2 pulls that were 2700 diamonds and 1 that was scrolls, which is when we realized the fact they said 3 pulls from each ‘pool’.

Franz
Franz
Reply to  Wattstrom
4 months ago

No bro, simple maths, when you pull 30 you’re closing the probabilities of having an elite, that does’t mean that you are going to have an guaranteed elite every 30, only if you open 220 you will obtain a number close to 10 elites. The probabilities are like that.

Dutchlion
Dutchlion
Reply to  Wattstrom
10 months ago

I also emailed. They said it’s a chance game and not every 3rd 10x summon you get an elite. I havent been getting any elite lately. 10x 10x summon. Count was 43 Bleu and 57 green. Only elite I got was with the card from the 100. So if you can get in writing theve said 1 in every 30 I’d like to see that because then I’m gonna stalk lilith games

Franz
Franz
Reply to  Wattstrom
4 months ago

That isn’t like it works, the number 30 only meaning that you’re closing to have an elite, you have not guaranteed, only if you open a number extremely big, like 3,000 or 30,000 summons you are going to have a number similar to the expected. For that reason sometimes you open 50 and don’t obtain anyting and later you obtain 3 elites in 10 pull, the probabilities only tell you wich is the number that you can expect for the number of tries. YOu can’t expect every worker in Liliths know about maths.

BlackBacon
BlackBacon
11 months ago

You’re vastly over-complicating basic statistics here. On average you should expect an elite hero every 21.69 pulls. (1/.0461)

I’d expect heads 50% of the time when flipping a coin. That doesn’t mean you should expect heads every flip.

Deaven
Deaven
Reply to  BlackBacon
11 months ago

Agreed. This just seems bad. Almost like Lilith is trying to make the pulls seem better than they are.

You definitely can’t expect an elite in 5 pulls 20% of the time.

Kuroi
Kuroi
Reply to  AFKArena
11 months ago

Yes the math is wrong, but it’s easy to see how you made the mistake. In probability, the expected value is given by multiplying the probability of each outcome by the outcome. So using the formula you had before, you would take P(n=1,k=1,x=4.61%)*1+P(2,1,4.61%)*2+P(3,1,4.61%)*3… all the way to infinity, and this would give you the expected number of tries needed to get an elite. Through some mathematical trickery (the formula for the sum of a geometric series), we find that this is the same as 1/4.61% = 21.69 pulls as stated above by BlackBacon

Kuroi
Kuroi
Reply to  AFKArena
11 months ago

Part of the reason you probably got confused is because you actually calculated the “median” rather than the expected value (aka the mean). Basically if you kept repeating an experiment it would take around half of the people 14 or fewer tries to pull at least one elite, and the other half 15 or more tries. But you could theoretically try, say, 100 times and not get an elite. If you take the average of how many tries it actually takes for people to pull an elite, a big number like 100 would drag the average number of tries down… Read more »

Salkasm
Salkasm
Reply to  Kuroi
11 months ago

Hmm I see what you mean, that makes sense to me. Ill look into it.

Salkasm
Salkasm
Reply to  Kuroi
11 months ago

Just to double-check: The formula for the calculation in the graph is correct, but the expectation part is wrong, I calced the median, rather than the expectation value?

BlackBacon
BlackBacon
Reply to  AFKArena
11 months ago

Your math is spot on. The “When to expect an elite hero?” section it is wrong though.

When flipping a coin 100 times you should expect 50 heads (50% odds).

When making 100 pulls you should expect 4.61 elites (4.61% odds)

Expecting an elite every 15 pulls means you’re expecting 6.66 elites in 100 pulls which would put the odds at 6.66%.

Salkasm
Salkasm
Reply to  BlackBacon
11 months ago

Ah, I understand it now. Alright thanks! Thats why I put the “not securely determined yet” there cause I wasnt 100% myself. But I see the mistake now, I will have it changed πŸ™‚

Salkasm
Salkasm
Reply to  BlackBacon
11 months ago

Hey there,
following your logic, I can make an example. Lets say you want to know how many times you need to roll a dice to get a 6 (which has a 16.67% probability, or 1/6). With the formula you used, that would be (1/(1/6))=6. The problem has been solved by mathematicians to be n=3.8 (or 4 when rounded upwards) rather than 6.

BlackBacon
BlackBacon
Reply to  Salkasm
11 months ago

You’re talking about something completely different. The claim here was “You should expect an elite every 16 pulls”. When rolling a dice you should expect a 6 every 6 throws. Every 100 throws you should expect 16.67 6’s. The larger your sample size the more accurate this becomes.

Salkasm
Salkasm
Reply to  BlackBacon
11 months ago

Yup I see that now. I calculated the median, which is not the amount to pull to expect a success. The reply was from earlier when I still got median and expectation mixed up.

Salkasm
Salkasm
Reply to  BlackBacon
11 months ago

The probability the dice problem has been calculated was by this formula: 1-(5/6)^x>0.5. It incorporates the 50% parameter, which your formula doesnt. Applied to the hero case, its 1-(1-0.0461)^x>0.5 which is about 14.6. x here is the amount of pulls.

The dice problem can be read online, as well as the procedure on how to calculate it.

Mark Tao
Mark Tao
11 months ago

In the early game ive got elite quite often, but now….I usually get 1 elite every 50-60 pulls.

Stephenie
Stephenie
Reply to  Mark Tao
11 months ago

Ive seen a 10 pull of all green heroes and the same has happened to my Guild Boss…I think their numbers are, just a bit, on the optimistic side.

Terpentin
Terpentin
Reply to  Mark Tao
11 months ago

Same here. Also it seems to me, if i open all 10 card at the same time with the button, i rarely get an Elite hero. If i open them one by one, i get more elites. Just a feeling.

Lappy
Lappy
Reply to  Mark Tao
11 months ago

I’m experiencing that too, it’s like they drop frequently to retain you then let you hit droughts once you’re established enough.

DarkHorse
DarkHorse
Reply to  Mark Tao
11 months ago

SAME! open a new account and you get a bunch of elite in the first 100% pulls. I have gotten 5 elites in my last 224 pulls. I know this cus i have not used my current tavern tokens and i am keeping count of the elites i have gotten. 3 of the elites are not on my wish list which is fully populated.