Probability for Wargamers 10 - roll 3, keep worst

Another popular game mechanic - roll M dice, keep the best or worst N. D&D players should be familiar with it, as roll 4 dice, keep 3 was one of the more popular ways of rolling stats. In this case, though, the discussion cropped up on the TFL list, in respect of artillery fire in They Couldn't Hit An Elephant, their set of divisional/corps level ACW rules.

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)

And our expected number of successes is 5/8 with an 1/8 chance of two. Interesting!

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.

blue = 2d6, green = 3d6l2