Trollslayers - Probabilities

Warning: math ahead

In Tunnels & Trolls, ability checks are determined by rolling 2 dice, with doubles re-rolling and adding to the result, then comparing that result to a difficulty number.  This will be the primary way of determining success (or failure) in Trollslayers for pretty much just about everything.  I think it would be beneficial to take a look at the probabilities that such a method produces.

First, by way of comparison, let's look at the odds of rolling 2 dice, straight up, without any additions from re-rolling doubles.  The number on the left is the dice result, the middle number is the percentage chance of that number coming up, and the number on the right is the percentage chance of that number, or a lower result, coming up:

2          2.78%          2.78%
3          5.56%          8.33%
4          8.33%        16.66% 
5        11.11%        27.77%
6        13.89%        41.66%
7        16.67%        58.33%
8        13.89%        72.22%
9        11.11%        83.33%
10        8.33%        91.66%
11        5.56%        97.22%
12        2.78%      100.00%

(yes, we're rounding fractions off, which makes the chart a little wonky - deal with it)

Now, let's look at the odds using the 'doubles re-roll and add' method, with up to two re-rolls figured (special thanks to Jasper Flick over on the RPG.net forums for pointing out how to figure this using his AnyDice program):

3          5.56%          5.56%
4          5.56%        11.11%
5        11.27%        22.38%
6        11.27%        33.64%
7        17.13%        50.78%
8        11.58%        62.36%
9        12.05%        74.42%
10        6.36%        80.77%
11        6.83%        87.61%
12        0.99%        88.59%
13        1.47%        90.06%
14        1.02%        91.08%
15        1.50%        92.59%
16        1.06%        93.65%
17        1.39%        95.04%
18        0.94%        95.98%
19        1.11%        97.09%
20        0.65%        97.74%
21        0.66%        98.40%
22        0.35%        98.75%
23        0.34%        99.08%
24        0.17%        99.25%
25        0.15%        99.41%
26        0.14%        99.54%
27        0.12%        99.66%
28        0.10%        99.76%
29        0.08%        99.83%
30        0.06%        99.89%
31        0.04%        99.94%
32        0.03%        99.97%
33        0.02%        99.98%
34        0.01%        99.99%

This isn't an exact replication of the odds, because in theory you could keep re-rolling infinitely.  In practice, it's close enough for government work.

One of the odd quirks of using the 'exploding' dice is that in many cases, a number has a slightly higher chance of occurring than the number below it, expectations to the contrary.  This is because even numbers are more likely to have a set of doubles create that number, which would then re-roll and add again, creating a larger number as a result.  It looks a bit weird, but as long as we're aware of it, it shouldn't trip us up.

It also gives us a few benchmarks to work with.  Because there's almost exactly a 50% chance of rolling a 7 or less, we can say that any roll with roughly 50-50 odds needs a 8 or greater to succeed.  As a result, eight is the base target number in Trollslayers.  Also, if you wanted to define something as having a 'one in a hundred' or 'one in a thousand' chance of occurring, you could assign target numbers of 23 and 30, respectively.

Also, we can figure that a +1 die modifier represents roughly a 12% chance in altering the odds, based on the percentage needed to alter an 'average' number of 7 (or an 8, based on the mean average - works either way).

I think this gives us plenty of material to chew on.  More to come later...