On Conga Lines and Triggering Traps

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    When exploring a dungeon, one of the ever present dangers is the potential existence of traps. One interesting bit of Old School Wisdom that I have discovered is that traps normally have a 2-in-6 or 1/3 chance of going off whenever someone passes by the trigger. 

    I personally think that this is a very neat feature, but it raises the question of determining who gets affected by a trap. One person? Just roll a d6 and look at the numbers. But what about when you have a long line of explorers? It isn't uncommon for adventuring parties to bring along lots of help or befriended monsters, so the number of people passing by a trap might be quite large indeed. Slowing down play to roll a d6 for every creature passing the trigger is inefficient, so I started looking into alternatives.

    Due to quirks of probability, while the first person in line will have a 1/3 chance of triggering a trap, the person behind them will only trigger the trap when 2/3 of the time the person in front of them didn't trigger it and they roll the 1/3 chance to trigger it, which can be expressed as 2/9. Effectively, every person has 2/3 the chance of triggering the trap that the first person did, so any nth position explorer has 2^(n-1) / 3^n odds of being the one to set off the trap.

    After calculating these odds, I was able to develop the following table:

# in Line d6 d12 d20 d100 Truth
1 2 4 7 33 33.3%
2 3 7 11 56 55.5%
3 4 8 14 70 70.3%
4 5 9 16 80 80.2%
5 - 10 17 87 87.2%
6 - 11 18 92 91.6%
7 - - 19 94 94.3%
8 - - - 96 96.2%

    To use this table, choose a die. For example, the d6. Roll the die and check the result with the table. The 1st person in line would be affected on a 2 or less (as is standard), and the second would be affected on a 3 then. With the d20, the 1st person would be affected on 7 or less and the second between 8 and 11. So on and so forth.

    This table has a couple glaring issues, unfortunately. First, is that obviously the die rolls do not map perfectly onto percentages. By the truth column, it can be seen that 80% of the time a trap will activate after 4 people have walked on it, but that leaves an additional 20% chance of someone else getting it. That would only happen by rolling a 6 on a d6, which isn't quite 20%. Also notable, is that as you move further back in the party, the more skewed the rolling odds get, particularly on the d6 and d12. 

    Finally, this table does not extend infinitely, and it is perfectly possible for a larger party size than this table to appear. Luckily, you are able to roll on this table again, just treating the people who haven't been potentially affected yet as a new line. For example, assume you have a party of 12 people. Just roll the d12 once and consult the table. If a 12 comes up, just roll again and treat the 7th person in line as number 1 in line, 8th as 2nd, and so on. If another 12 comes up, the entire party made it through without triggering the trap (low odds, but not impossible). 

    Hopefully this table is helpful to you all! Let me know if you find any issues with my math, and I will get it corrected as quickly as possible!

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