Thoughts on Math-hammer - tweaked for 8th Edition Relevance

Since 40k is a dice game, it probably pays to know a bit about how the dice will behave in a given situation.  8th edition shakes up a few things (like the to-wound chart and leadership) and changes some odds (like casting psyching powers, which is easier).  It also adds the whole 're-roll ones' mechanic, which tweaks re-rolls.

A Few Disclaimers - 
1) remember that we're talking about odds.  Averages are guides for decision-making.  You'll be able to take a guess at outcomes if you're weighing options.

2) Averages emerge when you throw dice enough times - twenty throws of a dice is about average, when it comes to statistics - IE - thirty guardsmen walk into rapid-fire range and drop 100d6 looking for 4+, you'll probably see 50 hits most of the time.

3) Remember that we have a cognitive bias towards remembering outliers - we're bad at math unless we're trying.

Basic Math - looking for X+ on 1d6

Base WS/BS	Unmodified chance	Re-roll misses	Re-Roll 1's	Base with +1 Bonus
2+	83%	97%	97%	83%
3+	67%	89%	78%	83%
4+	50%	75%	58%	67%
5+	33%	56%	39%	50%
6+	17%	31%	19%	33%

I'm going with WS/BS on these because it's the most common, but this math applies to any rolls when you need X or better, such as morale and wounds.  I've added in re-rolls, re-roll 1's, and +1 bonuses.

Note that blanket re-rolls are a bit better than a +1 bonus - but they're pretty close.  The only place that really loses out on it is the 2+, as we're assuming a 1 always fails here.

Note also that re-roll ones are neat, but have diminishing returns as you need better numbers when it comes to odds.  However, some stuff (like plasma) REALLY doesn't want to deal with 1's, because explosions.

Roll at Least X on 2d6

At least X on 2d6	Odds
2	100%
3	97%
4	92%
5	83%
6	72%
7	58%
8	42%
9	28%
10	17%
11	8%
12	3%

This one's important for two big phases - the psychic phase and the assault phase.  Two obvious break points are 9+ and 10+.  

We care about 9+ for deep-striking assault troops (which means that they've got just under a 1/3 chance of assaulting off deep-strike).  Note that any deep-striking troops that can get an extra inch of charge range (IE - Trygons with adrenal glands) get a significant boost to their ability to charge out of reserves.

We care about 10+ for Smite's bonus damage.

Roll 2d6, pick the highest

Roll 2d6, pick highest	1	2	3	4	5	6
Chance of getting at least  -	3%	8%	14%	19%	25%	31%

As long as I'm doing math, went ahead and did this one.  

So, What Do I do with this?

These are all guides - if you're looking at figuring out how nasty something's going to be, crunch numbers.  

Example 1 - 5 Chaos Terminators deep-strike and fire combi-bolters into guardsmen in cover. Let's assume they're all within 12" of the targets, so their rapid-fire 2 guns are kicking out four shots each.

5 combi bolters * 4 shots each = 20 shots

20 shots at BS 3+ = 13.33 hits on average

13.33 hits that need 3+ to wound (S4 vs T3) = 8.89 wounds

Guardsmen out of cover have a 5+ save

8.89 hits vs 5+ save = 2.96 saves, so 5.9 guardsmen die.

So you could expect 5-6 guardsmen to eat it.

Example 2 - say the Chaos Lord deep-strikes in behind his Terminators, and gives them the ability to re-roll 1's

5 combi bolters * 4 shots each = 20 shots

20 shots at BS3+ with a re-roll one 1's = 15.55 hits

15.55 hits needing 3+ to wound = 10.37 wounds

10.37 wounds vs a 5+ save = 3.4 saves, so 6.9 guardsmen die

So, if you dropped the terminator lord in behind his terminators, you could guess that it'd kill 6-7 guardsmen in that context.

To bring it further - so the guard survives, and however many survivors get to rapid-fire the terminators and then probably assault (move 6 + 2d6 charge = 80-90% chance of a charge going off).

What Math can do is tell you the odds of stuff happening with dice involved - but what math CAN'T do is tell you what your opponent is going to do - does the opponent turn around and try to kill the Terminators to a man now, denying you the ability to disrupt his backfield?  Does the opponent say 'eh, screw it' and let your terminators wander around his backfield and focus firepower elsewhere?

In Closing

Having an idea of how math works gives you a decision-making tool.  The dice can always give and take away, but if you know math then you can guess how rolls will go, and ask yourself how much risk you want to accept for a given course of action.