First Dive into Theory

04 Aug, 2025

A Little Bit of Followthrough

I threatened promised the other day to start diving into some poker stats, but I had put off any actual Wikipedia scrolling research until this weekend. Most of what I share here is just synthesized from a great page on Poker Probability, but my main goal is to put together some commentary of what this may mean for our game design.

My undergraduate background is in engineering and computational analysis, but I had a great pre-calc teacher in high school who taught us basic combinatorics. Combinatorics is basically the study of counting things, which is most of the battle when it comes to probability. If you're reading this as a fellow TTRPG hobbyist, the classic combinatorics problem for dice is rolling 2d6. There are 36 "ordered" or "Distinct" combinations, like you'd see in a d66 table, but there are only 11 possible sums of the two dice. You can count the Frequency of each sum and divide it by the Distinct Combinations to get the Probability of a given sum. So while it's a 1-in-36 chance of rolling a 6 THEN a 1, it's a 1-in-6 chance of rolling a total of 7. These are the basic concepts we can take with us to a playing card game, but we might have to get into a little more fussing with Combination/binomial coefficients to tell the whole story. Hopefully not!

Let's Ground Ourselves

Part of what made me nervous about actually crunching any numbers here is that I didn't have an intuitive feeling for how common poker hands are. If you've played poker for money, you know that conservative play tends to be king. I've only played hold 'em for cash, which is a variant where each player is dealt two cards and then, over subsequent rounds of bets, a total of five community cards are dealt to the center of the table. The player then assembles the best possible hand from the total of 7 cards they have access to (2 in hand, 5 in community). What I mean by conservative play, though, is that if your two cards in hand are total butt cheeks, you don't want to expose yourself to the risk of the betting rounds just to find out you might eventually have 1 pair. You would typically fold, and often you don't even see all of the cards for the round.

Because I play conservatively, I really only tend to see all 7 cards that a player had if they had a good enough hand to stay in. This is what I mean by a lack of intuition on how common some hands are. Balatro helps with this! By default, the player usually has 8 cards in their hand and Pairs are so weak and common that you're often throwing them away for a chance at, say Full House. So let's take a look at the table from our Poker Probability page and I'll append the relative score baselines from Balatro. Afterward I'll walk through how this affects my mental model of what a round of our game might look like. Additionally, if you're less familiar than I am with poker hands, this will act as our primer on the different ranking of hands. The probability column is for 7-card hands, like Hold 'Em.

Insights

I didn't expect this! You are more likely, in a seven card hand, to have a scoring hand than to have to play a high card. That's exciting, although a single pair is pretty overrepresented. If we return to our 2d6 comparison, we can make a very ugly probability density function correlation:

High Card (17%) ~~ Rolling 2, 3, or 4 (17%)
One Pair (44%) ~~ Rolling 5, 6, or 7 (42%)
Two Pair (24%) ~~ Rolling 8 or 10 (22%)
Three of a Kind, Straight, and Flush (total 12.5%) ~~ Rolling 9 (11%)
Full House (2.6%) is your highest ranking hand that occurs with any real frequency and it's about equivalent to your chance of rolling 12 (2.8%)

This is cool! I allowed enough slop in my comparisons here that we ignore any direct comparison to rolling an 11 (6% error is better than anything I've ever done at work), but we could practically run a PBTA game with this distribution. We could probably get a better comparison with 2d8 just because of the granularity but hopefully you understand my point - we now have a couple of frames of reference for relative probability density of certain types of hands if we linearize around 7-card poker.

When I say linearize, what I mean is that IF you have fewer cards-in-hand THEN you are more likely to have High Card/No Pair. Conversely, when you have more cards-in-hand you are more likely to have some valid hand to play. As a direct point of comparison, in 5-card poker (like some variants of stud) you have a 50.1% chance of having High Card/No Pair. Oddly, it's still a 42% chance of having a single pair, it's just higher-rank hands that drop off precipitously.

What does that mean for our game? Say there are abilities or effects that reduce the number of cards a character has in a round - you immediately clamp their ability to play something more valuable than a single pair. This calls to mind the Impair Gambit/condition in Mythic Bastionland. Characters typically have some pool of dice at their disposal during an Attack action; one die is selected to apply damage, and then any other dice that are a 4 or higher can be spent to apply Gambits. The Impair condition reduces the target's damage die to a 1d4; most results of a 1d4 roll will be blocked by a Knight's armor and only a 4 (25% chance) can be consumed to do something more useful.

Gamify It

When I started jotting notes down for how this system might work, I thought about successive ranks as representing either higher damage rolls or providing some sort of metacurrency that can be spent on things like gambits. The idea was that a High Card/No Pair hand could be spent on a basic attack that deals a weapon's base damage, a One Pair would provide a "tally" that could either be spent on a damage boost or a Gambit-style maneuver, Two Pair would provide two "tallies", and maybe there is some nonlinear function of hand rank-to-tally that would allow a player's turn to scale appropriately in effectiveness. Hence my comparison earlier -- if a character can Impair another character down to a 5-card hand, they've reduced that player's hand opportunity to a 50% chance of just dealing Base Damage and removed most chances to add damage or riders that turn.

As a followup from Pencil to Paper, I asked what the fungibility of redraws and hand size might be - I think today's notes have given me a little better intuition there. Overall hand size is still helpful in giving you the broadest set of options to play against, but discards let you be more intentional about hand composition - so the question ends up leaning harder against other rules in the set about riders/penalties for leftover cards in hand after a round, say.

What Might a Round Look Like?

Let's start with a simple initiative, I'm thinking simultaneous resolution for now.

1. Players (including the GM) draw [Hand Size] (HS) cards. Player HS is determined by their gear like we discussed before, GM HS is determined by a set of factors like in His Majesty the Worm; number of unique enemies, size of enemies, maybe unique riders on their stat block.
2. The players make decisions about which enemies they're targeting, and if two players target the same enemy, they can pool their hands to compose stronger hands - in-fiction this represents the PCs doing some cool combo. The GM probably has a much bigger hand in order to facilitate NPCs doing cool combos too. I'm a little worried that too many players pooling hands is going to cause issues of too many high-rank hands going off - the levers I see here are limiting initial hand size and allowing any number of players to team up, or having bigger individual hands but limiting combos to two players. Maybe the synthesis is that for each player in the combo, each involved player needs to burn a card from their hand to represent coordination and getting in position. I think allied NPCs would partner up with a Player Character and provide +X HS or +X RD
3. Players and the GM make decisions about [Discards/Redraws] (RD) in order to try to whip their hands into shape. I think it might be a little slower but more fun to do RD one card at a time rather than something like Balatro where a single Discard action is up to 5 cards at once. My model of RD is total number of cards that can be discarded per round.
4. Not sure about this step but I think this is when you would choose to burn cards in your hand to either perform miscellaneous actions or to "charge" your Armor - I like the idea of sacrificing action economy to actively use your defensive capabilities. It feels like it scratches a similar itch to Initiative in HMTW, where moving slower makes you harder to hit. Slow is smooth, smooth is fast.
5. Both sides reveal their hands and damage/riders resolve. Players and GM narrate their dope combos, everyone says "that was crazy", and we establish the stakes for the next round.

Stake Escalation

Josh from Rise Up Comus and I had a productive conversation last week where I shared an idea that's grown on me. Josh helped me to refine the idea some, and with what else we've talked about today I think it's got legs. In Hold 'Em, we mentioned earlier that the majority of a player's hand is actually in community cards. The first round of betting is on the draw (2 cards in each player's hands), the next round of betting is on the flop (first three community cards), next round is on the turn (fourth community card), and the last round is on the river (fifth community card).

I think this community card concept could add a very fun randomness element to our game. Player hand sizes are probably too big for 5 cards, but if we are designing for speed and decisiveness, I think starting with 1 card that anyone can use for their hand and progressively escalating each successive round would do a lot to show how combatants are really getting tangled up. My two gut caveats are that the community cards couldn't be used for miscellaneous actions or armor because they can't be discarded, and that they should probably be saved for bigger fights that should get more intense as they wear on. Maybe they represent an environmental effect or a ticking clock. Could be cool!

Completely Unstructured Thought on Magic

With the theme we've talked about, this is probably not a super-low-magic game. If someone is a magic user, I think it will look something like The Electrum Archive, where they know some countable number of spells that have varying costs and effects based on hand rank. Could be fun to write.

Let me know what you think!