The discussion (and someone from the list will no doubt correct me if I missed the main thrust of it) revolves around breaking up groups of guns into sections, and how unrealistically bad the resulting likely effects are. A four gun battery gets 2d6: the game mechanics for a two gun section (only getting 1d6) result in it being fundamentally ineffective against infantry in line.
[Aside: one preferred notation for 3d6 keep lowest 2 is 3d6l2, which I'll use from here on in.]
Let's look at the basic odds for the 'roll 3 hits, keep the worst two'. I don't think this is what TCHAE uses, but Napoleon at War is fond of similar mechanisms.
2d6: success on a 4: each dice has a 1/2 chance of a success, so:
- 1/4 chance of no successes
- 1/2 chance of one
- 1/4 chance of two
Expected number of successes = (1/4 x 0 + 1/2 x 1 + 1/4 x 2) = 1, probability of two successes 1/4,
1d6: successes on a 4: you have a 1/2 chance of one success, expected number of successes = 1/2, probability of two hits 0.
Fair enough. Now let's try 3d6 (you should have paid enough attention in the previous posts to work this one out!)
- 1/8 chance of 0
- 3/8 chance of 1
- 3/8 chance of 2
- 1/8 chance of 3
But what we actually want is to take the worst two, which becomes
- 1/2 chance of 0 (1/8 chance of 0 + 3/8 chance of 1 successs reducing to 0)
- 3/8 chance of 1 (2 successes reducing to 1)
- 1/8 chance of 2 (3 successes reducing to 2)
I'll leave the detail for success numbers of 5 and 6 to you.
More interestingly, perhaps - I suspect (not having seen the rules) that TCHAE actually uses a IABSM-style fire table, so in fact what we're interested in is a proper 3d6l2. I'll leave the detailed maths to you, but here's the table and graph of chance of rolling >= a given target on both 2d6 and 3d6l2.
Roll
|
>= on 2d6
|
>= on 3d6l2
|
2
|
100%
|
100%
|
3
|
97%
|
93%
|
4
|
92%
|
80%
|
5
|
83%
|
64%
|
6
|
72%
|
48%
|
7
|
58%
|
32%
|
8
|
42%
|
19%
|
9
|
28%
|
11%
|
10
|
17%
|
5%
|
11
|
8%
|
2%
|
12
|
3%
|
0%
|
(note, the probability of rolling 12 on 3d6l2 is actually just under 0.5%, hence rounding to 0%)
The interesting points - the 50% value moves from just over 7 to just under 6, but the odds on getting a 7 drop from 58% to almost half that - 32% - as the curve drops off quite steeply compared to the straight 2d6 roll.