Quick Notes on Blackrock Mountain Wing 4: Blackwing Lair

I’ve been noting down my experience with, and strategy for, the Hearthstone Adventure encounters each week on the EJ forum.  I just realized there’s no reason not to put them here for people working on the fights now.  I know they’re mostly over, but I’ll put in this week’s anyway, and maybe paste in the old ones sometime for people trying to do them later.

Descriptions of the bosses and their decks can be found here.

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Razorgore: Probably lots of easy ways to do this.  I went with typical Freeze Mage-ish board clears.  His only value advantage over you is creating 4-5 HP worth of stuff to beat down every turn, which is not hard to overcome just by establishing board control in conventional ways.  An early Doomsayer and a late Flamestrike help you lock it down though.  He Corruptions big threats, so typical midrange creatures to own the board work great.

Vael: Lots of ways to deal with the fact that he aggressively mills you.  I went with Rogue both for being able to spam out cheap cards right from the start (especially with Prep and Shadowstep), and being able to counter-mill him with Gang Up.  In my game, he milled out ahead of me–I guess he must have burned one of his Gang Ups–which made it easy.  If you ever get one good Blade Flurry off for a big reset, you should be fine.  He doesn’t have a huge number of minions in his deck.

BGH is handy for Giant spam near the end, and can be good with Shadowstep or Gang Up.  Other than that, you might not want much costing more than 2.  He will Naturalize creatures, so be careful trying to set up a huge threat like a Van Cleef.  But that can be great if you draw out the Naturalize (or see them burned).  Also, be very careful about lethal range, since he has 0-cost burn he can spam at you.

Chromaggus: Really fun encounter.  I think there’s a probably a great plan around using Warlock cards to discard from your hand, but I went with Priest.  Chromaggus has no hard removal, so a huge Divine Spirit-Inner Fire combo can win uncontested.  No matter what your deck, bias towards being very cheap again, since you have to spend a lot of mana clearing out his trash (the only exception for me was Thaurissan for obvious reasons).  I bet Cho is amusing if you have him.

You have to put with some of his curses for a while, and know when to get rid of them.  The initial Green will probably sit for a long time, so you have to first take control and then start damaging him.  I tried to use Lightwardens to abuse his auto-heal every turn (and to worth with my own Lightwell which was very good here).  You may have to live with the Red you get on turn 3 for a while also–you probably need to make a turn 3 play (especially since the turn 4 and 5 Blue/Bronze are really important to drop immediately).  Getting some semblance of control in turns 1-2-3 is huge.  You need some way of removing a growing Drakonid on your turn 3 (SW:P is good).

Bronze and Black should always go immediately (exception: consider leaving Bronze if you’re about to follow up with a good board clear).  Blue is ideal to let go of early on, since he can spam out 0-cost Flamehearts–but you can leave it if he’s already played a bunch of spells from his hand.  Drop Red when you can, and Green once you’re in control and ready to win.

Nefarian: Nothing about this fight seems to shout for one particular strategy–it’s mostly a normal Hearthstone game with him having a huge value advantage over you (starting at 9 mana/turn on turn 2, and getting 2 cards a turn).

You can probably use anything if you have a plan for the early game (just survive his 9-mana turns on your 3/4/5 mana turns), the mid game (probably need one really good board clear), and the late game (actually killing him and dealing with occasional damage).  I used Paladin with Humility/Chow/Doomsayer/Aldor for the early turns, an Equality-Consecrate whenever things got bad, and Lay on Hands/Guardian to take over at the end.  As with all the encounters where you’re overcoming a huge value disadvantage every turn, if you don’t have the typical good legendaries for that, a good endgame is to stick a Kel’Thuzad.  Nefarian also has no hard removal, but he does have Flamestrikes and various direct damage.

The random card you get from Rag on turn 3 matters a lot.  All you care about is living so the 6/6 Taunt seems best, especially if you can protect it for a bit and/or having him waste tempo removing it.  Mind Control Tech is huge, and Emperor Cobra can trade way up.  A Doomsayer can delay him a turn at worst.  Anything to get to the Equality-Consecrate or whatever you’re using to turn the corner.  Some luck with him doing comparatively weak things with his mana on turns 2-3-4 (such as using inefficient spells from your class) can also go a long way.

Quick Notes on Blackrock Mountain Wing 4: Blackwing Lair

I’ve been noting down my experience with, and strategy for, the Hearthstone Adventure encounters each week on the EJ forum.  I just realized there’s no reason not to put them here for people working on the fights now.  I know they’re mostly over, but I’ll put in this week’s anyway, and maybe paste in the old ones sometime for people trying to do them later.

Descriptions of the bosses and their decks can be found here.

————————————-

Razorgore: Probably lots of easy ways to do this.  I went with typical Freeze Mage-ish board clears.  His only value advantage over you is creating 4-5 HP worth of stuff to beat down every turn, which is not hard to overcome just by establishing board control in conventional ways.  An early Doomsayer and a late Flamestrike help you lock it down though.  He Corruptions big threats, so typical midrange creatures to own the board work great.

Vael: Lots of ways to deal with the fact that he aggressively mills you.  I went with Rogue both for being able to spam out cheap cards right from the start (especially with Prep and Shadowstep), and being able to counter-mill him with Gang Up.  In my game, he milled out ahead of me–I guess he must have burned one of his Gang Ups–which made it easy.  If you ever get one good Blade Flurry off for a big reset, you should be fine.  He doesn’t have a huge number of minions in his deck.

BGH is handy for Giant spam near the end, and can be good with Shadowstep or Gang Up.  Other than that, you might not want much costing more than 2.  He will Naturalize creatures, so be careful trying to set up a huge threat like a Van Cleef.  But that can be great if you draw out the Naturalize (or see them burned).  Also, be very careful about lethal range, since he has 0-cost burn he can spam at you.

Chromaggus: Really fun encounter.  I think there’s a probably a great plan around using Warlock cards to discard from your hand, but I went with Priest.  Chromaggus has no hard removal, so a huge Divine Spirit-Inner Fire combo can win uncontested.  No matter what your deck, bias towards being very cheap again, since you have to spend a lot of mana clearing out his trash (the only exception for me was Thaurissan for obvious reasons).  I bet Cho is amusing if you have him.

You have to put with some of his curses for a while, and know when to get rid of them.  The initial Green will probably sit for a long time, so you have to first take control and then start damaging him.  I tried to use Lightwardens to abuse his auto-heal every turn (and to worth with my own Lightwell which was very good here).  You may have to live with the Red you get on turn 3 for a while also–you probably need to make a turn 3 play (especially since the turn 4 and 5 Blue/Bronze are really important to drop immediately).  Getting some semblance of control in turns 1-2-3 is huge.  You need some way of removing a growing Drakonid on your turn 3 (SW:P is good).

Bronze and Black should always go immediately (exception: consider leaving Bronze if you’re about to follow up with a good board clear).  Blue is ideal to let go of early on, since he can spam out 0-cost Flamehearts–but you can leave it if he’s already played a bunch of spells from his hand.  Drop Red when you can, and Green once you’re in control and ready to win.

Nefarian: Nothing about this fight seems to shout for one particular strategy–it’s mostly a normal Hearthstone game with him having a huge value advantage over you (starting at 9 mana/turn on turn 2, and getting 2 cards a turn).

You can probably use anything if you have a plan for the early game (just survive his 9-mana turns on your 3/4/5 mana turns), the mid game (probably need one really good board clear), and the late game (actually killing him and dealing with occasional damage).  I used Paladin with Humility/Chow/Doomsayer/Aldor for the early turns, an Equality-Consecrate whenever things got bad, and Lay on Hands/Guardian to take over at the end.  As with all the encounters where you’re overcoming a huge value disadvantage every turn, if you don’t have the typical good legendaries for that, a good endgame is to stick a Kel’Thuzad.  Nefarian also has no hard removal, but he does have Flamestrikes and various direct damage.

The random card you get from Rag on turn 3 matters a lot.  All you care about is living so the 6/6 Taunt seems best, especially if you can protect it for a bit and/or having him waste tempo removing it.  Mind Control Tech is huge, and Emperor Cobra can trade way up.  A Doomsayer can delay him a turn at worst.  Anything to get to the Equality-Consecrate or whatever you’re using to turn the corner.  Some luck with him doing comparatively weak things with his mana on turns 2-3-4 (such as using inefficient spells from your class) can also go a long way.

From Dust to Dust: The Economy of Hearthstone

One of the most fascinating things about Hearthstone is that despite the usual terminology, it is not actually a “trading card game,” in that you cannot trade. Trading is functionally replaced by a crafting system that allows you to inefficiently transform cards into any other cards whenever you like. There are advantages and disadvantages to this from a player’s perspective. The obvious disadvantage is that you can’t shape your collection without destroying value. The advantage lies primarily in not exposing players to the vagaries of a secondary market as a requirement to managing their collections. This should be a big draw for people who have never played a TCG before, as every acquisition of a new card won’t involve the feeling that as a non-expert you might be getting cheated. Related is the topic of this post: since growing your Hearthstone collection is a solo endeavor, we can compute the rate at which it happens without reference to any market conditions or other exogenous factors.

Left out of the post is a discussion of Arena rewards and the efficiency of playing Arena. I want to add this to a follow-up post as soon as I have data on Arena rewards at each tier, particularly since there are a number of good reasons to spend your gold or money on Arena rather than buying packs.

For today, however, the question we explore is how many packs you must expect to buy or otherwise acquire in order to collect any desired set of cards.

The Facts

The key piece of data needed for this analysis is the average distribution of card rarity when you open a pack. For this I turned to any forum posts I could find where people tried to tabulate such information and made the best approximations I could: 1% Legendary, 4% Epic, 21% Rare, and the remainder common. 2% of commons are golden, and 5% of all other rarities.

  • I definitely would appreciate any further light people could shed on this in the form of card distribution data that was collected in a controlled manner (which means, it was non-selectively determined that a set of packs will be recorded). It would be a simple matter to tweak the numbers and re-run the simulations.
  • It’s not important for this purpose that each pack has a guaranteed rare or better. Since packs are only opened as whole units, only the average rarity across the pack is important. In theory the variance in the outcome may be slightly affected.

The other important information is the distribution of cards across each rarity. For this the database filters at Hearthhead come in handy. Select the “expert” set and leave “uncollectible” unchecked, and you can see the number of cards to be collected at each rarity:

  • 94 Common (40 neutral and 6 of every class)
  • 81 Rare (36 neutral and 5 of every class)
  • 37 Epic (10 neutral and 3 of every class)
  • 33 Legendary (24 neutral and 1 of every class)

Blizzard’s initial post on Arcane Dust is a reference for crafting and disenchanting values.

The Big Picture

From the above we can derive a few different things.

  • The total value of dust that would be required to craft a full playset (2 each of the 212 non-legendaries and 1 each of the 33 legendaries) is 106,120 Dust for a non-golden set and 428,800 Dust for a golden set.
  • The average yield of a fully disenchanted pack is 93.03 Dust.
  • Therefore, if you turned every card to Dust and crafted a full playset from scratch, it would take you 1141 packs for non-golden, and 4609 packs for golden. This provides a useful upper bound for comparison when we simulate how many packs it will take with more reasonable behavior.
  • With duplicates taken into account we would expect to open 135 legendaries before having at least one of each. At one legendary every 20 packs, that’s 2700 packs before you get them all by luck alone.

As always, before simply tossing everything into a simulation, it’s good to see what we can ballpark on our own. For example, we know that the expected number of packs opened to complete a set is going to be something substantially less than 1141. Let’s guesstimate that it will be less by a factor of around 2.5. We might think this because, if you imagine 106,120 Dust as that target value you’re trying to acquire, then the early cards you open will in reality contribute more than their disenchant value. They’ll effectively contribute their crafting value (because each one will be a card you don’t have to craft later). Since for the highest-value cards, the craft value is four times the disenchant value, we can think of the value of each opened pack as starting out around 4x the disenchant value and then decaying down to 1x as our collection fills up. So at an average of around 2.5, you might expect to open 450 packs in reality.

This assumes, incidentally, that you are using the optimal strategy for reaching the final goal, which is to disenchant all extras you open and hoard the Dust until you have enough to craft all missing cards all at once. Crafting earlier to reach other goals will be discussed below.

As one other interesting point of napkin math, with 450 packs you expect to open 22.5 legendaries. Obviously not enough to fill out the required 33, and in fact with overlaps you only expect to have about 16. So half of the legendaries will be obtained from crafting. This means we can conclude that crafting is generally going to go “up” (turning common cards into rare cards) and that around 27,000 Dust, almost 300 packs’ worth, will be put towards crafting legendaries to complete your set.

The Law of Large Numbers

The code used to generate the simulation results can be seen here. You can run it yourself in any Python interpreter, if you want to play with some of the options discussed below or make any changes of your own.

The first thing I simulated is the process described above: opening packs until you can obtain a complete playset. For this first run, the simulator disenchanted any card once it had more 2 non-golden and golden copies combined, and will always favor disenchanting golden copies (but the final collection may still have some golden cards in it). In other words, the most efficient approach to obtaining a complete set without regard for golden, and without doing any premature crafting.

The results were, over 10,000 runs:

  • The average number of packs needed was 512, with a standard deviation of 42.
  • The average amount of Dust used to craft cards at the end was 28,522, or 27% of the total craft value of the set.
  • 100% of commons, 99% of rares, 85% of epics, and 54% of legendaries were found by opening (rather than crafting).

There already a lot of meat for discussion here. To start, we finally have a number of packs we can expect to buy to finish a set. Depending on whether 512 is higher or lower than you expected, you might reevaluate your own goals an expectations based on this. Against a target of 51,200 gold to spend on packs, it’s also clear that the 40 per day from dailies is small potatoes. A full collection is going to involve buying a significant number of packs with cash (a fact that should be obvious if you consider how the game’s business model has to work). The desire for a full collection might be seen as an extreme case, but if we’re going to do this analysis at all, we may as well dispel the notion that you’re going to get there by doing less than a few years’ worth of dailies.

If you bought it all with cash, 512 packs would be $640 at the best bulk rate in the in-game store. Any daily quest for 40 coins will save 50 cents off of this, and any Play mode win will save 4.17 cents. In other words, a full card set can be valued at any proportionate mix of $640, 1280 daily quests, or 15630 Play mode wins.

Arenas will complicate this, and I expect will generally increase the efficiency. Your 150 gold investment in an Arena can be bifurcated into a 100 gold purchase of an ordinary pack (your guaranteed prize pack) and a 50 gold Arena fee. The 50 gold Arena fee will get you some mix of gold, Dust, packs, and golden cards, depending on how many games you win. I’ll avoid speculating on the value until I have data like I said, but observe that it would not take much for the mean prize value to exceed 50. To evaluate a Dust, remember that at the end of the process we turn a huge pile of Dust into all of the cards we don’t yet have. The last few packs of cards are likely to be turned mostly into Dust to meet this goal. Since a fully disenchanted pack is worth 93 Dust, acquiring 93 Dust earlier in the process willy likely save us one pack or 100 gold. So as a rule of thumb, 1 Dust will be worth around 1 gold. Coupled with the fact that an extra prize pack (100 gold) or a golden rare+ card (shaving anywhere from 100 to 3200 Dust off of the process) can, even at small probabilities, increase the expected prize by quite a bit.

Before moving on, don’t ignore the fact that the variance of the result is quite high. At something resembling a normal distribution, where 2.3% of samples exceed two standard devations in either direction, 2.3% of people will finish by 428 packs, and 2.3% won’t be done after 596 packs.

Immediate Gratification

All of the above was premised on the fact that you act so as to most efficiently reach the goal. In reality, you will likely be less efficient in a few ways:

  • You might craft cards before your collection is complete, when you need them for decks. This runs the risk of crafting a card you’d have eventually opened.
  • You might disenchant cards you don’t have 3 of, again for the purpose of immediate deck desires. This may cause you to re-craft the same card later.
  • You might purchase packs at less than the bulk rate of 40 at a time, incurring a higher per-pack cost.

On some level these actions can’t be evaluated in detail, as they depend mostly on your specific preferences regarding having cards sooner rather than later. We can estimate the downside however. For example, since in the simulations, 99.9999% of commons were opened in packs rather than crafted, you can assume than any crafting of a common is a waste of 35 Dust in the long run: the craft cost minus the 5 you’ll recover from disenchanting it later. The same is true even for rares–you have a 99% chance to be out 80 Dust at the end of the process. Whether these costs are worth it in any case is up to you, but it’s worth understanding why that Dust is a long-term cost to your collection.

Crafting legendaries is actually a better proposition, as there’s a 46% chance you wouldn’t have opened it even after 512 packs (note the sanity check here against our napkin math of half the legendaries opened after 450 packs). In that case, you incur no cost at all. If you do have to disenchant that same card later, however, it’s a waste of 1200 Dust.

Remember our earlier logical expectation that crafting should generally go upwards. It stands to reason that, early in the process, disenchanting common cards and crafting very rare cards is the safest thing to do, as it’s the most likely to be something you’d have had to do anyway. Finally, keep in mind that your odds of any crafting being wasteful will decrease as your collection fills out.

All That Glitters is Gold

For people interested in the full all-golden set, this is the same simulation over 10,000 runs:

  • The average number of packs has gone up to 2789, with a standard deviation of 101.
  • The average amount of Dust for crafting is 220,369, or 51.4% of the total value.
  • 77% of commons, 69% of rares, 35% of epics, and 19% of legendaries were opened in packs.

So, don’t aim for a full gold set unless it’s worth more to you than a new computer or two. In general you can interpret all of these numbers similarly as the non-golden numbers, but I want to highlight that the increase in pack requirements may be larger than you expected. The more expensive golden cards only require 4 times the Dust of their non-golden counterparts, but the commons you need are no longer easy to find, making the whole operation take substantially longer.

At any rate, these numbers are interesting just for the purpose of being careful with expectations. Goldens are best thought of as a fun bonus but, perhaps even more more so than in games which allow trading, collecting them is not a welcoming proposition. Specifically, in games which allow trading, the variation in people’s preferences allows the market to move foil cards from people who don’t care about them to people who do. Here, you’re on your own.

You Can’t Always Get What You Want

The most important practical refinement of all this is that most people will have goals smaller than a full set. The simulator is easily modified to work toward a “target” which is any subset of the full card set, if you give it the rarity distribution of the target set (for reference, the distribution in the full set is [94, 81, 37, 33]). In this case, it will open packs until the dust collected is sufficient to craft everything missing from the target set all at once. There is an option to either hold “unwanted” cards as usual, or automatically disenchant them.

I will include a few informative examples here. The value in parentheses is the result if you disenchant all cards outside the target subset.

  • No legendaries [94, 81, 37, 0]: 287 (248) packs. This seems like the first reasonable way of reducing the amount of cards you need.
  • 2 specific legendaries only [94, 81, 37, 2]: 306 (267) packs. If there are a couple general-use legendaries with widespread constructed application.
  • Full neutral set + one complete class [46, 41, 13, 25]: 425 (377) packs. Another reasonable goal, as it gives you full access to all possible decks with your favorite class. However, it’s still a lot of packs due to all the neutral legendaries.
  • Same as prior, with only the class legendary [46, 41, 13, 1]: 216 (145) packs.
  • Let’s say half the cards in the game are constructed playable [47, 41, 19, 17]: 389 (326) packs. Perhaps serious constructed players would get some value from going down the entire Expert card list and deciding the cards at each rarity they think are going to be usable, and running the sim with that breakdown.
  • You really want to go straight to some decklist you saw (here I turned one legendary into an epic since the sim currently collects cards in pairs) [5, 4, 2, 2]: 143 (57) packs. So a competitive netdeck with 2 legendaries can be obtained somewhat quickly if you’re willing to disenchant as needed to get it.
  • Downgrading the legendaries to epics [5, 4, 3, 0]: 104 (35) packs.

I’d be curious about any other examples people think are worth running, or run on their own.

From Dust Were Ye Made

For people who have played TCGs before, it can at first be hard to get our heads around the notion of card collection being a single-player experience. And even once you do, the math governing how long it takes to reach various collection goals is nonobvious and counterintuitive.

In particular, your progress towards completing a full set of cards is not even roughly proportional to the number of cards you have already. This not only because early packs give their value by adding cards you don’t have while late packs give value much less efficiently in the form of Dust, but also because a large portion of the value needed to complete the set lies a small number of rare cards. The legendary cards, even though they comprise 33 of the 457 cards in a complete set, account for 50% of the Dust value. And even this slightly understates their worth, as they are also disproportionately difficult to acquire naturally (i.e. despite the high Dust cost, the ultimately efficient use of Dust is mostly to craft legendaries). This is all to say, the best loose estimate of how far along your collection is is not how many cards you have, but how many legendaries you have and/or could craft with your current Dust.

It will be up to every player to decide how they prefer the single-player collection environment of Hearthstone to the market that defines most similar games. But the unique Hearthstone system was very ripe for a mathematical analysis of how it can be expected to play out. Much has been and will continue to be written on the gameplay of Hearthstone, but I hope I could improve our understanding of the other aspects of the game that will in large part shape our experiences playing it.

Hearthstone Probabilities and the Monty Hall Effect

The Monty Hall Problem

A certain probability puzzle is well-known in math circles for its unusual ability to cause people to refuse to believe the answer when it is explained to them. It’s usually known as the “Monty Hall problem” (after the host of Let’s Make a Deal):

Monty Hall has given you a choice of three identical doors. Behind one is a car and behind the other two are goats. You choose a door, but before it’s opened to reveal your prize, Monty adds a twist. He opens one of the other doors to reveal a goat (he always does this to add to the suspense). He then offers you the choice of staying with the door you chose, or switching to the remaining unopened door. Should you stay, or should you switch, and what’s your chance of winning in either case?

The answer is surprising to most: switching doubles your odds of winning the car (2/3 chance of winning, as opposed to 1/3 if you stay). The key fact is that Monty’s knowledge of which of the other doors (if any) was a car causes him to always remove a goat from the prize pool. The chance that the initial door you chose contained a car was 1/3 to start, and it’s unchanged by Monty’s ritual. But if the car was behind either of the other two (2/3 probability in total), Monty will remove the losing door and leave the winning one, and switching will win.

(If you don’t buy that the probability is anything other than 50% when everything started out equal and there are now two doors remaining, there are myriad sources on the internet trying to explain in different ways).

The Jaina Proudmoore Problem

Let’s say it’s late in the Hearthstone game and you’re about to try playing your bomb legendary. You’re pretty sure you’ll win if it sticks for a turn, but if it gets answered immediately you might be hopelessly behind. You want to guess whether your opponent Jaina is holding the Polymorph you know she has in her deck (assume for the moment she only has 1 for simplicity; the same concept applies with 2 but the math is more intricate). Her hand has 4 cards, and you mouse over her deck and see she has 16 cards left. After her draw next turn, 5 of 20 unseen cards will be in her hand, so you’d think she had a 25% chance holding the card.

Monty Hall, however, tells you differently. Even though she only has 5 cards in her hand next turn, she’s been selecting out non-Polymorph cards and playing them all game. Just as Monty selected out non-winning doors and removed the pool, making the remaining doors of the ones he could have chosen more likely to be winners, Jaina has been casting non-Polymorph cards from her hand, making the ones she’s left in her hand more likely to be Polymorphs.

For now consider the simplest case, where nothing had been cast so far this game that Jaina would have been likely to Polymorph. Applying the logic from above, there’s a 50% chance that the Poly started the game in the bottom 15 cards in the deck, and that probability has not been changed by any subsequent events. There must, then, be a 50% chance that it’s among the 5 cards in her hand. Quite a significant difference from the 25% that seemed completely intuitive before considering this effect.

Complexities

An important subtlety is that Jaina might not be as selective as Monty. Monty never prematurely revealed a car, but Jaina may have cast a Polymorph before the critical turn, if she had it and the game state called for it. The assumption above that the game state on prior turns never looked like one that would have drawn a Polymorph elides a deep point of probability, namely, the mysterious way that Jaina’s/Monty’s selectivity shifts probability between chosen and unchosen cards/doors. I think getting into the depths of how that happens might be beyond the scope of this post, but for now observe the following.

If you know Jaina is going to save her Poly for your bomb no matter what (and this might not be a bad assumption in a constructed card game where people are familiar with each other’s decks), then the situation is identical to Monty Hall: the probability in the above example would have been 50%. But if you made the opposite assumption, that Jaina is not smart and dispenses cards from her hand at random (ignore the vagaries of the mana curve for the moment), the probability would in fact be the 25% we naively estimated at the start. The best way to see the difference is to really understand why the answer to the Monty Hall problem is what it is. It might also be helpful to look at that bit of logic from the end of the previous section (about the 50% chance that Poly started in the bottom 15 cards of the deck) and try to see why it doesn’t apply in the case of the Jaina that plays randomly.

The other complexity is what I alluded to at the start: the arithmetic is more complicated with two Polymorphs in the deck (but as I said, the logic is unchanged). To work the same example with two Polymorphs:

  • The naive estimate is that out of 20C2 (20 choose 2, referring to combinations) places the Polys could be, 15C2 choices have them in the bottom 15, so the chance of having one in hand is 1-(15C2 / 20C2), which evaluates to 17/38, or around 45%.
  • The Monty-corrected estimate would be that out of 30C2 possible placements at the start of the game, 15C2 have them undrawn in the first 15 cards, so the chance of having one in hand is 1-(15C2 / 30C2), which evaluates to 22/29, or around 76%.

The result is the same as before, a much higher chance in the latter case.

Here is a full table of the probabilities for various deck and hand sizes, for reference:

Dropbox Link

Conclusion

The probability subtleties discussed at the beginning of the last section mean that guessing the probability that your opponent has a certain card is never an exact science. It depends on your judgment of how likely it would have been for them to use it at an earlier point in the game if they’d had it. The Monty Hall effect applies most strongly when it is highly unlikely or impossible (for example due to mana cost) that the card in question would have been played already. Where it applies, it causes the probability of the card being in your opponent’s hand to be substantially higher. So it’s both practical and perhaps fascinating to realize that you can’t rely on what you thought you knew about probability: all unseen cards are not equally likely to be the card you care about.