Weird thing about the chance one roll has to beat another:

A while ago, EastwoodDC posted a spreadsheet for calculating the chance of rolling any given result on a number of whatever sized dice. I started looking at the chances one roll had to beat another and I noticed something weird.

When rolling 1d6 against 2d6, the chance that they'll tie is equal to the chance of rolling exactly 7 on 3d6, and the chance that 1d6 will beat 2d6 is equal to the chance of rolling 6 or less on 3d6.

When rolling 2d6 against 3d6, the chance that they'll tie is equal to the chance of rolling exactly 14 on 5d6, and the chance that 2d6 will beat 3d6 is equal to the chance of rolling 13 or less on 5d6.

When rolling 3d6 against 4d6, the chance that they'll tie is equal to the chance of rolling exactly 21 on 7d6, and the chance that 3d6 will beat 4d6 is equal to the chance of rolling 20 or less on 7d6.

So I'm pretty sure that, anytime you roll Nd6 against (N+1)d6, the chance they'll tie is equal to the chance of rolling N*7 on (N+N+1)d6 and the chance that Nd6 will win is equal to the chance of rolling (N*7)-1 or less on (N+N+1)d6.

Messed around a little more, and it looks like 1d6 ties 3d6 as often as 4d6 will roll exactly 7; 1d6 beats 3d6 as often as 4d6 rolls 6 or less; 2d6 ties 4d6 as often as 6d6 will roll exactly 14; and 2d6 beats 4d6 as often as 6d6 rolls 13 or less. So I expect that, when rolling Nd6 against (N+2)d6, they'll tie as often as (N+N+2)d6 rolls exactly N*7, and that Nd6 will beat (N+2) as often as (N+N+2) rolls (N*7)-1 or less.

Wait. Those look the same. Can I just follow this pattern for any number of d6?

Nd6 ties (N+M)d6 as often as (N+N+M)d6 rolls exactly (7*N).
Nd6 beats (N+M)d6 as often as (N+N+M)d6 rolls (7*N)-1 or less.

And then, since the "7" there is a result of the average outcome of a single d6, can I extend it to other sizes of dice?

NdD ties (N+M)dD as often as (N+N+M)dD rolls exactly ((D+1)*N).
NdD beats (N+M)dD as often as (N+N+M)dD rolls ((D+1)*N)-1 or less.

Seriously just guessing at this point (could probably figure it out by hand with more spreadsheets, or by pure math from Pascal's Triangle) but I think this is going to work.


EDIT: VanVelding has tackled the math for this here and here.
EDIT ²: and here.

Final thoughts on TRO:3063

I was going to write more about TROs, but I don't have my notes together yet, so... TRO:3063!

It was initially released back around August 25th with the final version posted on September 9th. You can download it from The BattleTech Reader, OurBattleTech, BattleTechUniverse, Lords of the Battlefield, or the BattleTech Engineer. If you are considering a similar project, I highly recommend you look at the advice in Steve's 'Lessons Learned' document. Much of it applies to non-BattleTech, non-book projects as well.

There's already been some discussion of the TRO and the reception has been overwhelmingly positive. It matches, and in some ways exceeds, the quality of the current official TROs -- if I knew nothing about BattleTech, I don't know if I could tell the difference between it and Catalyst's books. (Granted, it was produced as a leisure activity over five years; Catalyst does not have that freedom.)

Looking at it now in retrospect:

Many readers seem curious about why the book covers the units it does or why they're built the way they are. (Power level, prevalence of SPLs & AMS, lack of infantry/spaceships/Clans, etc.) The reasons have been touched on in a few places, but for a book like this, it would be helpful to have a foreword explain more completely what the writers hoped to achieve and some of the thinking behind their decisions.

TR:3058, TR:3060 and TR:3067 show an overwhelming trend towards putting fusion and XL engines on new and upgraded vehicles. (The Tokugawa and Schiltron, for example.) I don't know that it was intentional, but TRO 3063 fits that trend quite well.

The manufacturing dates published by Catalyst (TechManual, Tactical Operations and so on) haven't been accurate enough for me to take them as Gospel, so me and Steve talked about how far R&D might've gotten by 3063. I suppose in-universe editorial comments [set off in square brackets, the way TR:2750 does] could have covered those weapons from a 3070s point of view.

I miss the Heavy PPCs. Not for the PPC itself, but for what having a heavy main gun forces you to do with the rest of your armament. It's more elegant. Less jumbled to read and fewer weapons to roll during play. (On the whole, the designs in this book are probably a little too optimized for my taste, but not moreso than what I remember of Catalyst's trends for the period.)

It's so hard to tell what's changed since I did my editing pass because I reviewed most entries only once, looked mostly for just the biggest issues, and tried to preserve/highlight (instead of replace) the original intent. (If you see a spot with too many three-clause sentences, or where paragraphs are all weirdly equal lengths, that's probably me.) And I really need to thank the other editor(s?) for going through entries after/before I'd gotten to 'em. But man, I'm scrolling through the book, and I know Steve's changed stuff, but it's so hard to pick out. I've only got two so far: looks like he overruled my cuts to the Feng-Niao, and I think he took a couple SPLs off the Sabra.