Dust Shop

## **Sold-out Shop**

It is possible to obtain every card in the game by spending Dust in the shops.
The Adventurer shop sells an incomplete set of cards and is relatively easier to buy out. The Expert shop sells every card in the game, including legendaries.

When each shop is completely sold out they show the following messages:

## **As of update 1.6.0 (hot update)**

ALL LEGENDARIES ARE NOW IN THE SHOP

Everything below this message is old and should be disregarded.

### **Confirmed Sold in Shop**

Note: This data is gathered from reliable players seeing a legendary appear in the shop. The data aslo includes banner announcements.

### **Chance to see a Legend in the Expert Shop**

#### **Confidence Interval Calculation for This Chance**

CI = xbar **±** Z(alpha/2) **×** sigma / **√** (n)

* Where xbar is the mean
* Z(alpha/2) is 1.96 for a 95% CI
* sigma is Standard Dev
* n is sample size

in place of sigma / **√** (n), we can use **√** (( p\*(1-p) ) / n) where p is the measured percentage of the event measured.

We can do this because it is a proportion. Based on a normal approximation of the discrete binomial distribution that yields under the following conditions: (1) n\*p and n\*(1-p) are both be greater than 5, (2) and n is greater than 30.

**Calculations:**

CI = 0.25 **±** 1.96 **×** **√** ((0.25**×**(1-0.25))/131) <- (this assumes p is 25% with a sample size of 131)

CI = 0.25 **±** 0.0742

CI = 17.6% to 32.45%

**When the Expert Shop is refreshed, the chance to see a legendary card is somewhere between 17.6% and 32.45%.**

Further data is needed to narrow that margin.

#### **Not Sold in Shop**

Many in the player community believe these legendaries are NOT SOLD in the dust shop

* Anabel, Dawn's Light
* Benny, Silver Tongue
* Bill&Billy
* Burian Iron Roar
* Dazzling Angel
* Eisele Bloody Soul
* Immortal Ficus
* Marsa, Blackgold Rose
* Meson 3rd
* Powley, Petrel
* Reinhardt, Militant
* Ruma
* Soar, Fleuret Gent
* Solar Dragon
* Yiyun Sward Master Ren

### **Shop Status unknown**

These cards were in the shop during early versions of the game. They have not been seen in the shop for several months. They are probably not currently offered in the shop.

* Ancient Earthshaker
* Bloodeye Tyrant
* Caroline, Holy Lance
* Leira, Raging Flame
* Master of Cosmos
* Mekki
* Peripatetic Lay Monk
* Pigsy (last seen 11/17)

#### **Probability to get a specific legendary in the shop**

* **IF** All legendaries are sold and all have the same ratio:
	+ 1/44 = 2.3% but a legendary only shows up every 25% so 1/176 = 0.57%
	+ About 175 refreshes on average should allow the specific legendary to show up
* **IF** Only confirmed legendaries are sold and all have the same ratio:
	+ 1/21 = 4.76% but a legendary only shows up every 25% so 1/84 = 1.19%
	+ about 84 refreshes on average should allow the specific legendary to show up
* **IF** Only confirmed legendaries are in the shop, but some appear more often than others (using v1.5 data only):
	+ Example Istorvir has a current frequency of 1, so 1/(48\*4) = 1/192 = 0.52% (48 is total sample size)
	+ Istorvir needs about 192 refreshes. Dinin No.9 has a current frequency of 5, and thus needs about 38 refreshes

### **Confidence Interval that all legendaries show up in the shop**

If the conventional wisdom is incorrect and all legendaries are actually sold in the shop AND with the same ratio, there is a 0.57% chance to get Caroline. This means you should see her in about 175 refreshes. Is this correct?

We can use the RULE OF THREE to determine the likelihood of an event that does not occur. If an event does not occur in *n* times, its probability is less than 3/*n* with 95% confidence.

https://www.johndcook.com/blog/2010/03/30/statistical-rule-of-three/

**Example**

* You have seen 50 legendary sightings, but never Caroline. What is the probability that she will show up?
* Rule of 3 says that her probability is 3/50, or 6%.
* If we noted 200 legendary sightings, her probability would drop to 1.5%

This IS the 95% interval. This means that in the example above, the 95% confidence interval says the chance of seeing Caroline is between 0% and 6%, or 0% and 1.5%.

Note: this is not including the "no legendary chance". If a legendary shows up every 25% of the time, you would expect to see Caroline every 267 refreshes (using the n = 200 and probability is 1.5%)

**How many is enough?**

 With a current sample size (*n*) of 55, the % range of Caroline showing up in the shop is pretty wide.
It would be nice to have a definite % chance but that is impossible--the sample size would have to be enormous. And with each game version we separate the data just in case the devs change the list of available legendaries. That reduces our available sample size for our calculations.

* To get a CI of 0% to 1%, n must be 300
* To get a CI of 0% to 0.1%, n must be 3,000

**One more thought to consider**

Many players know Caroline is OP. A lot of players are watching for her in the shop! Players do not talk when they see someone like Dinin, but if they saw Caroline they would tell everyone.
No-one in the Discord has reported a Caroline sighting since 2017.
The fact that no one has seen her implies that *n* is larger than the records in the table. It would include every shop refresh observed by every players in the Discord as well. The absence of any report suggests there is probably no chance to find her in the shop.
This can never be proven but is a very compelling argument for the hypothesis that she is not in the shop OR that her chances to appear are extremely low.

### Epics

* v1.5
* n/a
* Farseer Witch
* Hungry White Dragon
* Chief Bigmouth
* Fearless Warrior
* n/a
* Frozen Memory
* Deus Herald
* Ink-House Hermit
* Reynard Ronin
* n/a
* Thundercloud Summoner
* Forge Master
* Petrel's Bodyguard