One Year of Thronebreaker Daily Crystals (sort of)

So it has been a year and change since the Thronebreaker title arrived, and with it the Thronebreaker Daily Crystal. I've been recording every crystal opening to measure the drop odds for this crystal. It is a little late as I've been busy, but here's the first 365 days of TB Daily crystals.

In 365 days I opened 326 crystals. I didn't actually miss any days directly (I logged in every day). Rather, because the 24 hour timer restarts when you claim the crystal, the next one arrives later and later during the day until eventually one day it happens after I go to sleep, whereupon I effectively "lose" a crystal. 326 crystals means I'm only getting about 89% of them, and letting daily wrap around cost me a crystal about once every nine days or so. I could do better if I set an alarm or something, but I have no intention of doing that.

The one drop I have yet to get: 10k 6* shards. I just keep striking out there. But also curious: the last time I got T5CC fragments was on November 30, 2020. I've had three T5CC drops, all within the first two months. I appear to have been either super lucky initially, or super unlucky after that, or the odds have changed. Possibly all three.

Given my opening rate, factoring in the fact that I don't open one per day (I lose a few here and there), on average how much stuff have I been getting in an average 30 day month?

those 30 day averages are total amounts, not drops. In other words, I've been averaging about 6.4 T4B catalysts per month. That's not bad for free crystals, and competitive with other "free" sources of resources, such as the Thronebreaker calendar.

Assuming I'm seeing something close to the actual drop odds for the crystals, I would say the value in this crystals is appropriate for Thronebreaker freebees. The value is high enough to be interesting, every drop is at least useful, but the value isn't so high as to be game altering. It strikes a good balance between being valuable enough to be worth getting, but not so valuable that it is painful to lose one or two.

At this point I'm waiting to see how long before the 10k 6* shards. Is it a 500 to 1 drop, a 1000 to 1 drop, or am I just cursed. Hopefully I'll find out while 6* champs are still relevant to the game.

Trivia:

5* champs pulled to date: Korg, Dragon Man, Vulture

I've gotten the exact same drop three times in a row several times. I have yet to have the same drop four times in a row.

I've gotten at least one full T4CC of every class. The class I've had the most full catalyst drops, by far, is mutant (five). Skill, Science, and Tech are all tied at one.

If a player decided to claim these over time and then open them in bulk, they could open a year's worth of crystals in about twenty minutes total (between claiming and opening). A year's worth got me 90 units, which would have been 270 units per hour of time spent (had I not spent all this time logging everything). That's one of the best unit returns per minute you can get in the game (of course, it is unsustainable).

Comments

Squirrelguy

Here’s some more data for you. Still awaiting the free 6* or even the T5CC, but I’ve done pretty well otherwise.

Crcrcrc

Hey, at least your percentage is better than the 65% 18k t4b I've gotten since I got tb

MavRCK_

There is a 10k 6* prize. Rich got it

RichTheMan

i most certainly did.

Malreck04

i most certainly did.

Mr flex himself has graced our presence

TIERA_10203

is there compensation for cav daily?

DNA3000

i most certainly did.

Every time I spin a daily and it is not the 10k drop, I will tell myself that at least Rich got it, and this will make me feel much better.

rockykoston

All I want to know is do you prefer opening in batches of 5 or 10 or more? Or a single crystal everyday?

Qfury

Not gonna lie.. every time I open a daily now and it's not 6* shards I'm severely disappointed.

I understand why my dad left and never came back now. I feel the same type of disappointment daily.

unknown

This content has been removed.

Menkent

There is a 10k 6* prize. Rich got it

I got that from my second ever TB daily... never seen it since

Mord

So I just saw in reddit somebody else posting a full year results of TB daily crystals (credit to Merek2445), have a slow day at the office, and though maybe calculate the probabilities for this two samples together.

But... why would it be interesting? after all they won't change all that much, right?.

Well, let's give it a different perspective that is not always examined: I will be calculating the uncertainties or parameter estimation intervals as well.

Because there are 15 different outcomes, some of those with seemingly low probabilities, 1 year worth of data for 1 or two players won't give a very accurate frequentist mean estimate. Also, frequentist confidence intervals are typically not well understood or interpreted, so I'm going to use Bayesian statistics instead.

Method:

We have 15 different outcomes, each with probability P_i, where sum(P_i) = 1. This follows a multinomial probability distribution.

P(y|p) = likelihood = multinomial probability distribution

It returns the probability of obtaining Y_i outcomes counts, given the set of probabilities p_i

The conjugate prior of the multinomial distribution is a Dirichlet distribution. If we know nothing a priori about the potential outcomes probabilities, we can build a Prior following a Dirichlet distribution with parameters alpha_i = 1. This will set a prior with uniform density over all outcomes (p_i =1/15)

If we then follow Bayes rule, the posterior:

P(p|y) ∝ P(y|p) P(p)

Will return the probability distribution of each p_i, given the data we collect. If we pick a D(alpha_i = 1) as Prior, this posterior will be a Dirichlet distribution with parameters Y_i (simply the counts per outcome). If you want details about this you can check for example A. Gelman's 'Bayesian Data Analysis' book.

To calculate the high probability density intervals, where we have 95% probability of finding the actual parameter value *given the data*, I will simply grab a large sample from the posterior and calculate the 2.5 and 97.5 percentiles. There will be a lot of resources to sample from a Dirichlet, but I'm using R because... well... R is awesome.

Below I show both samples, y1 and y2, the total count y, the 95% probability interval low/high bounds (%), the prior and posterior means (also %) and the odds calculated from the posterior mean:

As simple picture, I'm showing below the sample posterior distribution for the probability of pulling 18k t4b shards:

This 95% HDI (high density interval) represents that we have a 95% probability of having p(18k t4b shards) between 16.4% (low) and 22.2% (high). The width of the interval is the uncertainty due to not having a (super!!) large sample.

And that's all!. For anyone asking, yes, I like to overcomplicate things for no good reason.

Squirrelguy

Thought I would combine even further for possibly more accurate final numbers on the percent chances (and to compare/contrast). Still salty that I have yet to pull T5CC or 6* shards......

*Also, disclaimer, but I have a horrible memory and very occasionally I forget what I pulled from the daily when I go to log it. On the bright side, I can guarantee you that it is because it was not a memorable pull (like a full 6*) and was most likely one of the more common pulls which impacts the overall statistics and accuracy far less.*

Here is the link if anyone wants to edit and add their own for a greater sum of data:
https://docs.google.com/spreadsheets/d/1v6luULCYbBTLEWSclilL8i83uqnDuPk-vdNF1pGoIx0/edit?usp=sharing

Mord

@Squirrelguy

I was combining the samples from DNA3000 and Merek2445, you can remove my column as it is repeating DNA's sample.

Kudos to anyone actually saving the results, I don't have the will to record it daily!

Squirrelguy

Lol clearly I'm not awake yet. Thanks @Mord, sorry all.

DNA3000

If you want my latest live results to aggregate, here they are:

Mord

Just in case somebody is interested, here's another D(y_total_i + alpha_i) posterior sample results with HDI ranges. Remember I'm using alpha_i = 1 for non informative prior.

y1 = DNA3000
y2 = Merek2445
y3 = Squirrelguy

We can see how the intervals shrink as the sample grow.