You've got questions? We don't blame you... our Quick Tables seem too good to be true!
How do I use the tables?
We will use the 8d4 distribution as an example. Simply start at Table 1 and roll your d60. Look up your d60 roll in the first row of the table and your result will be in the row directly below it. Most of the time it will be a number and that is it - you are done! Your result is mathematically identical to if you had rolled 8 d4s and summed them up. Sometimes the result is not a number but a downward facing arrow. This means that you need to go to the next table and repeat the process.
How is this even possible?
Let's use a very specific distribution to make it easier to understand:
A has a probability of 1800 / 3600 (50%)
B has a probability of 1799 / 3600 (about 49.9722%)
C has a probability of 1 / 3600 (about 0.0277%)
Now here are the d60 Quick Tables for that distribution:
In table 1, a d60 roll from 1-30 results in A, 31-59 results in B and a roll of 60 means you have to roll again using table 2. So 30/60ths (50%) of the time the result is A, 29/60ths of the time the result is B and 1/60th of the time you roll again using table 2.
In table 2: a d60 roll from 1-59 results in B and rolling a 60 results in C.
The chances of getting outcome A are pretty intuitive: 30/60 = 50%
So what are the chances of ending up with outcome C? To get C you need to first roll 1 of 60 faces on table 1, then 1 of 60 faces in table 2. So there is a 1/60 * 1/60 = 1/3600 chance of outcome C.
Finally, what are the chances of getting outcome B?
29/60 in table 1, plus
(1/60 * 59/60) = 59/3600 in table 2
Since 29/60 = 1740/3600, the chances of getting outcome B are 1799/3600.
Using just two tables we expressed a very specific distribution where the rarest outcome only had a 1 in 3600 chance of happening. Yet, 59 times out of 60 we would get our result in the first roll. If 1/60 table 2 led to a third table, we could have shown a result with a probability of 1 in 216,000. Each table lets us show more rare outcomes and also allows us to fine tune our results as we saw with outcome B in the example.
With this technique, we can recreate any distribution as accurately as we want to, no matter how complex it is.
Is it really perfectly accurate?
Yes, except in the vary rare case that you end up in the final table where the probabilities are within a few ten thousandths of the target distribution at the most. We note the precision below the last table in each distribution. We could calculate as many tables as we want for an arbitrary precise fit, but at some point that becomes impractical so we choose a stopping point and very slightly round the probabilities of the distributions.
This rounding only occurs in the final table, so any result from any other table is a perfectly identical match to our target distribution.