A few weeks ago, there was a thread asking people how they defined success in MTGPQ. Multiple people answered that it was when they would collect all the cards, or something similar. Right around that time, I was also wondering how long it would take to collect all the cards. Having a mathematically inclined background, I decided to calculate that which would be hardest, but still remain reasonable: what is the cost to acquire all mythics within a given set?
Apart from the various assumptions made, this post is intended to contain only facts. (Only this post, not this thread, feel free to express your opinions, obviously.) I may give my opinion in a subsequent post. In any case, I hope the info presented will guide you in your card and app purchases. Also, all calculations made use probabilities and averages, and are intended to represent the average numbers/ amounts/ costs/ etc. Individual experience will inevitably vary.
NOTE: I am not a mathematician, I don’t know everything and this post is lengthy and math heavy, so I probably made some mistakes. Feel free to point them out if that is indeed the case. I also tried to explain as much as I can without giving a probabilities course, but I may have skipped some important stuff, so feel free to ask questions as well.
Assumptions
These are not necessarily true or reflective of actual game experience, but are made either to enable or simplify answering the question.
-
At the “start”, all other cards below mythic rarity are owned and 0 mythics are owned. This is to assume maximum booster crafting currency (bcc) acquisition. In reality, this will not be the case, and the cost to acquire all mythics will therefore be higher than what is calculated in this post.
-
Cards are only acquired through premium pack purchases and booster crafting, and crystals come from buying them for $ in the shop at the max amount (3000). This is mainly to be able to have a monetary value for accomplishing the goal. Also this is the only instantly and infinitely repeatable way of acquiring cards, even if it’s not the most cost efficient.
-
All cards within a certain rarity are equiprobable. Not sure if that’s a word, but it means that once a mythic is obtained, the probability of it being any of the possible mythics is the same. For example, if there are 26 mythics in a set, the probability of getting each mythic (once a mythic is obtained in a pack or through crafting) is 1/26. There is no way to know for sure if that is indeed the case unless D3 tells us, and is the best hypothesis to work with imho.
-
Each card acquisition is a completely separate event. That means that the probability of getting a card of a certain rarity, or a certain card within a rarity, is not influenced by and of your previously acquired cards. Once again hard to know for sure, but the most simple scenario to analyze.
-
Masterpieces are always assumed to be new cards. Depending on the set, the average amount of masterpieces acquired when all mythics are acquired will be around 1-2. I think this is a fair assumption. In any case, it wouldn’t change the conclusions very much and require more math than I think it’s worth.
-
Booster crafting is done only when there is enough accumulated currency to buy all remaining mythics. This is pretty obvious, but for those who haven’t figure it out: when you buy a pack you risk getting duplicates. The more cards you have, the higher the chance of getting duplicates. Crafting guarantees new cards. Therefore you should buy packs first and craft second.
The math
I’ve calculated the cost for a few sets of the regular format (I started this before M19). Here are the accompanying spreadsheets:
Color code: orange means input dependent on set, and green means final results.
Spreadsheet explanation:
-
BCC
This section contains the booster crafting currency cost and rewards for each card rarity. -
Pack “1 card” probabilities
This section contains the probabilities of getting at least one card of a certain rarity in a pack (shown in game when clicking the % icon on a pack). See next section for more details. -
Card probabilities
This sections contains the probability of a single card being of a given rarity in a booster (which I’ll call p). The game gives the probability (which I’ll call P) of getting at least one card of a given rarity (when you click the % icon on a pack) within a pack. This means that the probability of not getting at least one card of that rarity is 1 minus that number, so (1-P). This happens when not getting that rarity happens n times in a row, where n is the number of cards in that pack. The probability of not getting a card of a certain rarity on a single draw is (1-p). The probability of not getting it n times is (1-p)(1-p)(1-p)*… n times, so (1-p)^n.
So we have 1-P=(1-p)^n and therefore p=1-(1-P)^(1/n).
I’ve calculated this for every type of pack. Because of the precision of the given P, sometimes the values of p differ from type to type, so I’ve made a sort of rounded guess as to what p probably is give the calculated values. I figure the probabilities are probably rational numbers with max 4 decimals. Obviously I’ve made sure that the sum of all rarity probabilities is 1. Hour of Devastation is the worst one here, I’ve had to fudge the numbers a bit so it makes mathematical sense.
-
Average bcc reward
The expected bcc reward value of a duplicate for a certain card rarity is the probability of getting that card rarity times the bcc value of that rarity. Assuming all cards are duplicates, the average reward for a single card is the sum of all rarities’ expected rewards. The bcc from the bonus rare is also included here, averaged on 25 cards. Obviously, not all cards are duplicates. The mythics will sometimes be new cards. This is accounted for in the “cumulated bcc” column in the following section. -
Main table
This is where the cost of each mythic is calculated. Each line represents the acquisition of a mythic through pack purchases. The average number of cards required for acquiring the next new mythic is calculated (mean nb of cards for next mythic). The total amount of cards acquired is accumulated in cumulated cards at next mythic. Then, the cumulated bcc from those packs is converted in mythics (nb of mythics buyable with bcc) using the average bcc reward per card, and that bcc is converted in mythics giving the total mythics acquired + acquirable. The cumulated cards are also converted in premium packs, which are converted in crystals, which are converted in Canadian dollars. Finally, that crystal cost is divided by the total acquired mythics for that line.
Since the goal is to acquire all mythics, I’ve highlighted the lines where the mythics acquired + acquirable go from below to above the total number of mythics in the set. The actual average cost is somewhere between those 2 lines.
-
Mean nb of mythics before new one
When acquiring a mythic, there is a certain probability that it will be old or new. Since I’ve assumed that all cards within a certain rarity were equiprobable, the probability that it is old is the number of mythics already owned over the total number of mythics. The probability that it will be new is p=(total number-number owned)/total number. The average number of mythics n obtained to get 1 new mythic is n=1/p. (from n*p=1). P changes when getting a new mythic, which is why each line of the table represents getting a new mythic. -
mean nb of cards for next mythic
There are two ways of seeing this. It is either 1/(probability of getting mythic * probability of getting new mythic), or average nb of cards per mythic*average nb of mythic per new mythic. Both give the same thing obviously. -
Second degree model cost @ total mythics in set
Since the total mythics acquired + acquirable don’t necessarily arrive at the total number of mythics in the set, the total crystal cost actually lies somewhere in between the value of the line before and the one after. So I’ve fitted a 2nd degree polynomial to the data to find the actual average crystal cost. (Not included in the spreadsheets.) The fits weren’t too bad, and I didn’t feel like doing error analysis. I just approximated the error based on the difference between the data and the model. Feel free to improve this if it’s not enough for you.
Results and Conclusion
Average cost of acquiring all mythics within standard (pre-M19) sets.
Set |
Average crystal cost |
Average $CAN cost |
|---|---|---|
Origins |
21600 ± 100 |
1008.00 |
HoD |
19200 ± 100 |
896.00 |
Ixalan |
24500 ± 100 |
1,143.33 |
Rtl |
14780 ± 50 |
689.73 |
Dom |
21700 ± 100 |
1012.67 |
Once again, please remember that these are averages and that personal experience will vary, for better or for worse. Also, as you all know, there are many more ways of acquiring both cards and crystals, so the real cost, when considering freely acquired resources, should actually be less. Maybe this post will have inspired one of you to build a time cost dependent model using in game rewards, or maybe even a combination of both rewards+purchases. Also interesting would be a proper analysis of the cost variance. The average is a good start, but what are the probabilities of diverging from that average by a certain amount? Finally, whether these costs are reasonable or not is something we probably all have an opinion on, and I would love to hear what you think!
So that’s it! I hope you enjoyed this post and that the info contained will help guide you in your decisions.
