Hi there and welcome to the first installment of Horror by the Numbers, a series where I take a look at the numbers and probabilities behind some cards from Arkham Horror: The Living Card Game. As with most card games, Arkham has some amount of intrinsic randomness that comes from a randomized deck at the start of every scenario. In addition to that, it also has the extra special element that makes this a Lovecraftian style game, the chaos bag. We’ve seen plenty of cards that interact with this and plenty of amazing articles that analyze probabilities behind pulling a token. The objective of this series is to look at cards that go beyond pulling a single token and that would require some more complicated math to figure out. We’ll still be taking a look at cards that involve some simpler math, as those are a good baseline when comparing probabilities of other effects.
To start off, I’ll let you in on what tools I’ll be using for these numbers. The first has been the most used for me for most card games and it’s the Hypergeometric calculator. This tool is very handy for figuring out how likely it is you’ll see a specific result when you pull elements from a random set. You can use it to analyze how likely it is you’ll see a card in your opening hand or how likely it’ll be that you pull a specific token when pulling from the chaos bag. I’ll leave you to find out how to use more specific features of the calculator (or let me know if you’d like an article about it). Another great tool that’s a sort of hypergeometric calculator is Arkham Bag Calculator, but it lets you visualize odds of passing much better, as well as allowing you to see and customize what’s in the bag. Lastly, I’ll be using some code I wrote up to simulate some more intricate cards, especially ones that involve decision making. This code (written in C#, accessible here) will usually simulate a given situation 100,000 times to spit out a probability, usually with a margin of error below 1%. I’ll go over any finer points of the simulation when talking about specific cards.
Today, I’ll be focusing specifically on a mechanic that shook up the odds not only for cards in its cycle, but for cards all over Arkham, the Bless/Curse mechanic introduced in The Innsmouth Conspiracy. We’ve been used to some token shenanigans with the Seal mechanic, introduced in The Forgotten Age as well as cards like Grotesque Statue and Olive McBride. Now, we have a different way to mess with the chaos bag, by adding two new tokens, Bless and Curse. Some cards even reward you for revealing tokens of those types and we’ll take a look at some of those today.
Important note: For the purposes of simplicity, all cards will be analyzed with the Night of the Zealot chaos bag, which contains 16 tokens. Some campaigns start off with more, some with less, so results will vary depending on that. We’ll also assume none of the non-bless, non-curse tokens cause you to pull another token, but some scenarios can have that effect as well.
Simple, yet important
Let’s start off this installment by analyzing a few cards that are mostly modeled using simpler tools, but that are still worth looking at. We’ll start off with Ward of Radiance, which is usually compared to an old favorite, Ward of Protection. While there are many differences worth analyzing, the one we’ll focus on today is the reliability. How much can you count on Ward of Radiance? The answer depends on how many bless tokens are in the bag, and below is a graph showing you that relationship.
The first takeaway from this graph is finding the number of bless tokens you’ll need to give yourself the chance you want. In my opinion, 4 tokens is plenty good, giving you an ~80% chance at cancelling the treachery. Another point is that at high token numbers, you see lower returns on the probability. The difference between playing it with 6 tokens and with 10 is only around 6%. This means you probably don’t need to hold out for higher tokens numbers to pull the trigger on the ward, especially if the revealed treachery is dangerous enough.
Next up, we’ll take a look at a similar card, set to release very soon in In Too Deep, it’s Gaze of Ouraxsh. In this case, the result is not binary, so instead I’ll be looking at the expected damage value of Gaze given the number of curse tokens in the bag. This basically means, how much damage you’ll deal on average given that token composition.
Looking at this chart we can see that in general, we’ll be expected to deal 3 damage with half the curse tokens in the bag. Taking a closer look at that midpoint, we’ll see that we’ll do 3 or more damage about 69% of the time, leaving 31% of pulls with only 2 or just 1 damage. At a fully cursed bag, we’ll have a 65% chance of dealing 4 or more damage. The main takeaway from this card is that it’s not very consistent given its variant result, especially if you’re looking to deal more than 4 damage. That being said, it can be a good way to chip away at a boss enemy, rolling the dice to get lucky with your pulls. Like most token pulling shenanigans though, Olive McBride can alter this significantly, giving you a 92% chance of doing 4 or more damage and 72% chance of doing 5 or more with a fully cursed bag.
The card that should not be
Now we come to the featured card of this article, one that was heavily looked down upon when compared to the remaining members of its family, Paradoxical Covenant. I’ll preface this whole discussion by saying this card is very hard to trigger by simply pulling from the bag. Luckily to help our odds a bit, we have a few cards in the Mystic card pool that help. For the purposes of triggering the covenant, we’ll be taking a look at a few strategies outlined below and simulated using the framework I set up. I’ll plot the probability of Paradoxical Covenant triggering based on how many of each bless and curse tokens are in the bag, each increasing by a 1, always keeping an equal amount. Numbers will vary differently if we manipulate blesses and curses independently, but we won’t dive that deep into this card… for now. Here are the strategies:
Simple Pull: Just pulling from the bag. Nothing special.
Jackie Pull: Using Jacqueline Fine to pull extra tokens, but only after a bless or curse has already been revealed. After pulling one of those, we activate her ability, either choosing two non-autofail tokens or choosing the opposite token to what was already revelated. If neither of those are present, prioritize choosing another token of the type first revealed, to maximize additional pulls. For example, if we initially pulled a curse and activate Jacqueline’s ability revealing [curse, -1, skull], we’ll choose the curse hoping to find a bless off of that.
Olive McBride Pull: Using Olive McBride on the first pull, we choose two tokens prioritizing getting a matching pair. If we can’t, we choose as many bless or curse as possible to maximize additional pulls. For example. if we pull [-1, curse, curse], we’ll choose both the curses.
From the data, we can see that Olive is by far the best enabler for the covenant. As expected, simply pulling from the bag yields very low chances, staying close to 0% for low token counts and barely reaching above 20% with a full bag. Our strategy with Jackie is not that much better, reaching around 44% with a full bag and only 25% at half full. Since Jackie can usually only choose 1 out of 3 tokens, using her for the first pull doesn’t really help our chances and to activate her according to our strategy, we’ll need to first pull a bless or curse. At a full bag, we have a 55% chance of drawing one of those tokens and given that, after we draw said token, Jackie would give us an 80% chance of drawing its counterpart. Not bad, if we want to avoid failing because of one or multiple drawn curses. Olive gives us great chances from the start, especially at a full bag with over 80% chance to succeed. This is definitely the way to go if you’re looking to activate the paradox as often as possible.
The biggest question I’ve seen about these numbers is, how many of those activations actually matter? Unfortunately, that is a question I can’t answer with math or simulations. The numbers I’ve given are merely probabilities and they may affect your decision on when to activate Olive/Jackie or even when whether to take Paradoxical Covenant in the first place. Not every deck will be looking to take this, but given the cursed spell suite we’ve seen in The Innsmouth Conspiracy and the possible crippling effects of drawing multiple curses during a test, I think Paradoxical Covenant can be better than what most people give it credit for as a potential safety net for a the variance of a full bag.
Feeling lucky, Wan?
Lastly, I want to go over yet another card that has been much maligned by the community since its release. As the top review on his ArkhamDB page shows, Henry Wan hasn’t really been worth the effort. Until now? This card is very interesting because token pulling shenanigans aren’t as good with his effect, especially cards like Olive McBride and Dark Prophecy. It’s also interesting because the condition of success relies entirely on the player. You decide whether to stop gambling or not. Of course, stopping at 1 success will give you the highest chance of not busting, but you’re barely getting an action’s worth of value. For the Aspiring Actor’s ability, we’ll take first take a similar approach to Paradoxical Outcome and look at the probabilities of a number of strategies varying on the amount of bless and curse tokens in the bag. Here, the strategy will be to stop at a set amount of successes and we’ll calculate the chances of achieving that many successes. We’ll also raise the total number of bless/curse tokens in the bag since we don’t care what we reveal outside of the four symbols.
From the graph, we can see the odds of getting one success from our betting increases pretty well, topping out at 86% with a full bag. Where it gets interesting is at higher numbers. With the two pull strategy, we go from 46% with no tokens to 74% with a full bag. Given that, if we play Henry Wan and activate him 4 times, we’ll have spent 5 actions and be expected to have gotten about 6 resources/draws. Is this good enough to play? I’ll leave that up to you! To wrap up, I’ll leave you with a graph of the expected value for the number of pulls before busting for versus the number of bless and curse tokens in the bag. Notably, this counts the pull you busted on, so usually you’ll want to pull one less time than indicated.
That’s it for this inaugural installment of Horror by the Numbers. The Innsmouth Conspiracy is still releasing, so look forward to another installment when we get some higher XP bless/curse cards later in the cycle. If there’s any other cards you’d like me to analyze, let me know. Otherwise, feel free to send any comments, thoughts or concerns my way through Facebook, Discord or Nightgaunt Mail. See you next time!