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Encounters with Poissons and Exponentials
Posted By Matthew J. Neagley On June 19, 2012 @ 1:21 am In Tools for GMs | 9 Comments
The other day, Martin posed a question to me. To paraphrase: “How do I set up a die roll to determine how many encounters I have per day and when those encounters occur?” After some discussion, I suggested the below system, which is based on the Exponential distribution. Since we’ve gotten requests for info on this distribution before and the result turned out pretty neat, I wanted to share.
The exponential distribution isn’t a concept that exists in a vacuum. Instead it’s a function of the Poisson distribution, which is itself a function of the Binomial distribution, which is a function of the Bernoulli distribution. Here’s the rundown:
Thus, given that we know λ, our desired average number of encounters per day, if we make the reasonable assumption that our encounters follow a Poisson distribution, then the time, also in days, until our next encounter follows an exponential distribution. That makes determining exactly when the next encounter happens a fairly simple matter (with some spreadsheet help) and because the exponential distribution formula is based on the parameter λ, the average number of encounters per day will be spot on in the long term while retaining a minute chance of running into three encounters in an hour or none in a month.
To translate our formula into a die roll, we’ll need to use a table. In addition, because there are an infinite number of values the time until our next encounter could take, we have to use ranges of values instead of exact time values. To do this, we’d select the die we want to work with and our average number of encounters per day, and plug them into our probability formula for the exponential distribution, then solve for time. Here’s an example:
We want to use a d10. Each number on a d10 is 10%, a proportion of .1, and we want an average number of encounters per day of 1/6th. Thus our formula is .1 = 1-e-(1/6)x. Solving for x, we get –6*ln(.9) = x ~ .6 of a day, or 15 hours. This means that there is a 1 in 10 chance that our next encounter happens within 15 hours. Using this setup, we get results, but the average is too high because we’re breaking up the possible outcomes into chunks and using the greatest result from each. To get a result with a better average, we can adjust the proportions by –.04 each. This gives the following results:
|Time||9h||1d 1h||1d 19h||2d 16h||3d 17h||4d 22h||6d 11h||8d 14h||11d 19h||19d 8h|
This process works for other average encounter numbers, and the adjustment stays the same as long as you use a d10 (d20 uses a –.02 adjustment instead.) Here are results for a variety of average encounters per day:
|Ave. \ Roll||1||2||3||4||5||6||7||8||9||10|
|1/8||12h||1d 9h||2d 10h||3d 14h||4d 22h||6d 14h||8d 15h||11d 10h||15d 17h||27d 18h|
|1/6||9h||1d 1h||1d 19h||2d 16h||3d 17h||4d 22h||6d 11h||8d 14h||11d 19h||19d 8h|
|1/4||6h||17h||1d 5h||1d 19h||2d 11h||3d 7h||4d 8h||5d 17h||7d 21h||12d 21h|
|1/2||3h||8h||14h||21h||1d 6h||1d 15h||2d 4h||2d 21h||3d 22h||6d 11h|
|1||1h||4h||7h||11h||15h||20h||1d 2h||1d 10h||1d 23h||3d 5h|
|1 1/2||1h||3h||5h||7h||10h||13h||17h||23h||1d 7h||2d 4h|
The same process also works on other scales (such as a 10 min scale for use in high threat zones like dungeons) as long as you keep the average encounters and time in the same units. Here are some examples using the 10 min scale:
|Ave. \ Roll||1||2||3||4||5||6||7||8||9||10|
|1/8||5m||14m||24m||36m||49m||1h 6m||1h 26m||1h 54m||2h 37m||4h 18m|
|1/6||4m||10m||18m||27m||37m||49m||1h 5m||1h 26m||1h 58m||3h 13m|
|1/4||2m||7m||12m||18m||25m||33m||43m||57m||1h 19m||2h 9m|
With a table like these (feel free to grab and use these if you like) you can roll on the line for the average number of encounters that matches the area your players are currently exploring, and shift up or down rows to adjust for player behavior that may make encounters more or less likely. This allows for a great variety in encounter timings, with minimal rolling that actually matches real world statistics, which as we all know, is one of the most important requirements for a fantasy game .
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