### 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 Pulls | Common | Rare | Elite |
---|---|---|---|

At least 1 in 5 | 97.37% | 94.34% | 21.03% |

At least 1 in 10 | 99.94% | 99.69% | 37.63% |

3 x 2700 = 1 hero elite ???

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.

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

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.

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 »

That is not how it works, mate.

Dude you slept deeply during math class…

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.

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)

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’.

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.

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.

It’s based on the drop rates listed in-game. Are you saying the math is wrong? If so, I’ll gladly change it.

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

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 »

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

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?

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%.

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

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.

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.

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.

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.

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

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.

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.

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

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.