Five Solutions to the Birthday Paradox in Tables
“Has this ever happened to you?”, the infomercial MC asks threateningly while pots and pans clang, and elderly people slip and fall in the background. If you use random tables, particularly of the random encounter variety, you know the travesty of rolling the same result twice. This is incredibly common, especially with your standard 2d6 table, where a “7” result comes up 1-in-6 times. The best tables are written with this in mind with the most common results being the most repeatable, but some are not. But even a 1d100 table, which seems to offer a dizzying array of diverse results, repeats are sure to occur with frequent use.
This is the Birthday Paradox, the mathematically maddening marvel that you’re just as likely as not to find twinsies in a room of 25. Sure, I feel special to have the same birthday as Hodag, but maybe it’s inevitable given there are at least 25 other bloggers. As the Blog of Holding noted of the Birthday Paradox and its application to roleplaying games, “On a d20 chart, you can expect to roll a duplicate after only 6 rolls. On a d100 chart, you need only 13 rolls before you’re more likely than not to roll a duplicate.” There has to be a better way, the infomercial MC says with a smirk.
1. Fudge the Roll
You could just lie. You roll a result you’ve gotten before, oh well just pick a different result that catches your eye. Why not? What does it matter? Why does anything matter? Why are you rolling dice at all, you nihilist? Although this avoids the Birthday Paradox in its entirety, why did you roll? In my view (articulated in passing long ago in one of my least read posts of all time), if a result from a random table calls out to you, just pick it. Don’t roll at all. Rolling is for when you want to disclaim your decision-making to the plastic oracles, which you can’t truly do if you’ve made a decision. Once you do roll, you should commit to it. This is one of those age-old discourses and there is no argument in its favor (or counterargument) I could present here that would be even somewhat new.
2. Reroll
This is a perfectly adequate solution in a pinch. Keep rolling the dice until you get something new, like you’re resetting your Gameboy until you get the perfect natured, shiny Pokémon of your dreams. However, this feels terribly inelegant for the same reason that rolling to hit and rolling for damage does. You’re rolling multiple times for one result. I am just never a big fan of this and avoid it if I can.
3. Accordion Table
Now we are getting to the clever solutions. The first time I recall randomly encountering the accordion table (named because it expands as needed) in use was in the Horrendous Hounds of Hendenburgh adventure. It had a 1d12 random encounter table with the guidance that “If you rolled an entry already, use the one below”, and offered encounters numbered 1-15. This does two cool things: it eliminates the Birthday Paradox while giving encounters that can’t be encountered immediately. To have the 15 result occur, you will need to roll 12 three times (or 11 and 12 each twice, or other similar combinations). This gives the encounter table and the area overall a feeling of depth akin to that in a depthcrawl. It’s great and simple random table tech.
4. Overflow Table
You can drastically reduce repeats if you have two tables instead of one. If you roll a duplicate on your first table, then roll on the second table instead. This works with even very small tables. For instance, if the basic table is 1d6 and a second 1d6 overflow table, then it's only 12 separate encounters but exponentially fewer Birthday Paradoxes than with a 1d12 table, which is the exact same number of encounters. You can also design it so that the regular table has encounters you would rather not repeat, and the overflow table has encounters you don’t mind repeating at all.
This was a tool we used to great effect in Barkeep on the Borderlands. For those unfamiliar with the structure of that adventure, it is highly dependent on random encounters (the adventure has 100 random tables with 745 total results, not counting sub-results) whether you are traveling between pubs or fucking around inside said pubs. A few of the pubs, however, had only a few sidetrack encounters (i.e., encounters outside and on the way to the pubs) with 2d4 roll tables. There was the bright idea (I would love to claim credit, but the records are lost and my memory is too hazy, so it was either myself, Ty, Nick or Ava) to have overflow tables. There are four neighborhoods in the city the adventure is set in, so Ty wrote 1d6 overflow tables for each of these neighborhoods with a direction that these could be used as additional encounters when repeated rolls of sidetracks that had been exhausted. The tables both reduce the Birthday Paradoxes in the adventure and also gives some illustration of the character of the neighborhoods aside from which pubs are located therein.
I also built in overflow results in some of the more common results. For instance, most of the 7 results on the 2d6 tables had sub-results that meant that it wasn’t quite the same encounter each time it was rolled. To give a particularly silly example from Someone’s Apartment (a particularly silly pub), the most common random encounter is actually 6 different encounters written as rhyming couplets: Roll 1d6: (1) a widow enters, seeking a spouse; (2) a ratcatcher enters, seeking a mouse, (3) a wizard enters, too drunk to read, (4) 2d6 halflings enter, smoking pipe weed, (5) a priest enters, saying a prayer, (6) a patron exits, pursued by a bear.
5. Trust the Process
Variety is not always the spice of life. I actually like it when an encounter repeats because it tells me something. Sure, if the “wolves attack” encounter is a 7 on a 2d6 table, the second wolf attack certainly signals to players that this forest is chockfull of wolves, but even the less frequent encounters can be fun when prompted twice. They don’t play out exactly the same the second time, and it can allow the encounters to grow into beloved recurring characters.
For instance, when running Barkeep I had the following encounter come up twice while the player characters were at Our Lady of the Sacred Speakeasy: “A charming stranger tries to dance with a jolly crewmate. The stranger bites when close; they are a vampire.” The first time it was rolled, it went about as you would expect (the player character tried to flirt back and ended up being led to a spire on the pub’s rooftop for a private conversation). On the second roll, because the vampire was still upstairs with one player character, I took it to mean that there were two vampires out on the town. It turned out that they were actually a couple who had just broken up, amicably, and were both seeking rebounds but caused each vampire to become jealous of the player character that was subject to their ex’s newfound affections. Hijinks ensued, of course. If the first player had spurned the vampire’s advance, I may have run it differently with there still being one vampire who just moved on to their next target, seeming desperate.
Also, don’t sleep on the randomized number appearing roll embedded in some results. Luke Gearing has a post on how interesting 2d20 bandits is as an encounter. Rolling 40 bandits is very different (and tells you something different about the world) than rolling a pair of bandits, but both are possible. I like to play with this concept from time to time. As a recent example, in my Xmas dungeon, Castle Claus, the upper levels has an elf hostel where 2d6 elves are sleeping at any given time. However, I specify that if the result is a 2, then elves that are sleeping are specifically sleeping together and are quite annoyed to be interrupted. Number appearing rolls embedded within a random table immediately makes the encounter more varied and more repeatable without much effort.
This post was written for the Random Blogwagon. I rolled a 10, a 7 and pulled an Ace of Spades. If you want to participate but find the “Assigned Posting Window” table no longer works well because half of the results have passed, simply roll 1d10+10 on the table instead. Or keep rolling until you find a result that doesn’t require time travel. Or do whatever, roll tables are tools, not taskmasters.
