Dice Math, Exploding Dice VS +1 modifiers, and ReRoll Calculations in 40k 8e

I was looking for a way to compare Expert Marksmen and Bolter Drill, because I love my Sternguard and wanted to see what is better.

Here is what I found. +1 to wound is: count the normal number of faces that wound, divide the normal sum of unsaved wounds by that count; multiply that by the new number of faces that wound. So shooting Cultists (no armor save do to -2AP) is like this; normally 4 faces wound and now 5 faces wound: (10 * (2/3) * (4/6) * (1)) * 5/4

Shooting Chaos Marines is like this: (10 * (2/3) * (3/6) * (2/3))* 4/3. And for a Sanity Check, the number that is produced is identical to this formula: (10 * (2/3) * (4/6) * (2/3)) wounding Chaos Marines on 3+ due to +1 to wound.

This is ideally a quick way to see the impact of +1 to a die roll -- to get the multiplier you take the new number of faces that succeed divided by the old number of faces that succeed. This formula also applies to +1 to hit the exact same as wound. Shooting a Chaos Marine with +1 to hit reads like this:

(10 * 4/6 * 3/6 * 2/3) * 5/4. For all BS3+ models, if +1 to hit is the only modifier that equates to 5/4 more unsaved wounds.

If you stack +1 to Wound with +1 to Hit, VS Chaos Marines you get a formula like this, (10 * 4/6 * 3/6 * 2/3) * 5/4 * 4/3

Reading left to right that is Number of Shots times the odds to hit times the odds to wound times the odds they fail a save times the +1 to hit modifier times the +1 to wound modifier.

Bolter Drill, and Exploding D6s: The Multiplier can be summarized as, take the sum of normally expected successes and multiply it by the number of faces on the die plus one for each exploding face, and divide that by the number of faces.  

Exploding on 6+s means a multiplier of 7/6: 6 faces and +1 for 1 exploding face, divided by the number of faces.  

Exploding on 5+s means a multiplier of 8/6: 6 faces and +2 for 2 exploding faces, divided by the number of faces.

Exploding on 5+s for Crimson Fists also means there already exists the +1 to hit, for a total multiplier of (5/4 * 4/3), totaling 5/3 which is getting towards double the unsaved wound production.

Since this is so Dicey, here's Reroll Ones: Multiply the sum of unsaved wounds by 7/6.

Chapter Master reroll misses, when applies to BS3+ models is a multiplier of 8/6. The more difficult way is to say total hits is two thirds plus two thirds of the one third that initially missed. But 8/6 is easier to work with. This amounts to taking the normal expected total (6/6 of the total) and expecting two more die faces out of it (6 + 2)/6. This strategy applies to reroll failed wounds too, imagine you wound on a 4+. The multiplier is (6 + 3)/6. Reroll a 6+ to wound? Multiplier is 11/6.

Sanity check: 

• 1/2 + (1/2 * 1/2) == 1/2 * 9/6 == 0.75
•  2/3 + (1/3 * 2/3) == 2/3 * 8/6 == 16/18 == 0.888888
• 1/6 + (1/6 * 5/6) == 1/6 * 11/6 == 11/36 == 0.305555

So as for Bolter Drill VS Expert Marksmen it depends on the enemy's toughness. Assuming you're Exploding on 5+s due to +1 to hit, Bolter Drill is better against units of Toughness 7 and less. The multiplier is 5/3 VS 3/2. Against Toughness 8, Expert Marksmen pull ahead, 2/1 VS 5/3.

With exploding 6+s, (and no + 1 to hit), Bolter Drill is a multiplier of 7/6. Expert Marksmen is better VS T3 and above -- a multiplier of 5/4 at T3, 4/3 at T4, 3/2 for T5-7, 2/1 for T8. Don't waste a CP on Expert Marksmen VS T2!

So lets put all these multipliers together with 20 Sternguard Shots and see how many Ork Boys we can kill in one volley, with all the Buffs.

20 shots * 4/6 bs * 3/6 to wound * 6/6 armor save * 7/6 lieutenant reroll * 8/6 Kantor reroll * 8/6 Bolter Drill * 4/3 Expert Marksmen's +1 to Wound * 5/4 from +1 to Hit comes to 23 unsaved Wounds.

If, instead of the Lieutenant, you used the new Anti-Ork stratagem to reroll failed wound rolls, you get a series of multipliers like this: 20 * 4/6 * 3/6 * 6/6 * 9/6 from reroll failed wound rolls * 8/6 * 8/6 * 4/3 * 5/4 coming to 29 unsaved wounds.

And last, if you used no Multipliers, 20 shots * 4/6 BS * 3/6 to wound * 1 no armor save = 7 unsaved wounds. For 3 CP and a Chapter Master you can do 22 more wounds.