Probability and Expected Values of CP Farming

[Diagramdude]

You should be very familiar with the command point farming setup of Grand Strategist + Veritas Vitae in which you roll N+1 dice for an N cost stratagem, and that the Grand Strategist is worded "refunded" while the Veritas Vitae is worded "gain." Therefore on an optimal refund roll you can profit by 1 CP.

 

I'd like to reveal then the refund probabilities and expected values of a 1, 2, and 3 CP stratagem when you are running that CP regeneration setup.

 

Consider this 6x6 table that lists the possibilities for rolling 2 dice for the refund on a 1 CP stratagem:

 

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

 

Out of the 36 outcomes, on 16 you lose the CP (no 5+), on 16 you refund one and wash (a single 5+), and on 4 you gain a CP (both 5+). 

 

Therefore, when you spend 1 CP, you have a 44.4% or 4/9th's  chance to lose it, a 44.4% or 4/9th's chance to get it back, and an 11.1% or 1/9th chance to profit one CP. 

 

The expected value or expected CP cost then is 0.333 or 1/3rd of a CP. In other words, statistically, 1 CP cost stratagems actually cost 1/3rd of a CP! If you started with 10 CP, you could expect it to last 30 1 CP stratagems.

 

Now for 2 CP's, the process is similar but with three digits, 111, 112 all the way to 665 and 666. Imagine six of the tables above, each with a different leading digit 1 to 6. The first one would look like this:

 

111 112 113 114 115 116

121 122 123 124 125 126

131 132 133 134 135 136

141 142 143 144 145 146

151 152 153 154 155 156

161 162 163 164 165 166

 

And the sixth one would be:

 

611 612 613 614 615 616

621 622 623 624 625 626

631 632 633 634 635 636

641 642 643 644 645 646

651 652 653 654 655 656

661 662 663 664 665 666

 

Out of the 216 outcomes, in 64 you lose both, in 96 you lose one and refund one, in 48 you refund both and wash, and in 8 you profit one.

 

When you spend 2 CP, you have a 29.6% chance to lose both (64/216), a 44.4% chance to lose one (96/216), 22.2% chance to wash (48/216), and 3.7% chance to profit one (8/216). One huge takeaway here is that when you spend 2 CP, you have a 70.4% chance to get at least one back.

 

The expected value for a 2 CP stratagem is exactly 1 CP. So with the CP farm, you have a 50% discount on 2 CP stratagems!

 

The same process can be applied to look at 3 CP stratagems by adding a fourth leading digit. Sparing the details, out of the 1296 outcomes, in 256 you lose three CP, in 512 you lose 2 CP, in 384 you lose 1 CP, in 128 you wash, and in 16 you profit one.

 

When you spend 3 CP, you have a 19.8% chance to lose 3 CP, a 39.5% chance to lose two, a 29.6% chance to lose one, a 9.9% chance to wash, and a 1.2% chance to profit one. You have an 81.2% chance to get at least one back.

 

The expected value for a 3 CP stratagem is 1.67 or 5/3rds of a CP.

 

I find these probabilities valuable to know because I often find myself in a critical situation where I am running low on CPs and need to gauge whether it is worth it to use a stratagem when, if i blow the refund rolls, I won't have enough CP for another crucial stratagem. For example, if I have 4 CP left and I Honor the Chapter, if i don't make any of the refunds I will be down to 1 CP and can't afford any more 2 CP stratagems.

 

So, to recap:

 

Spend 1 CP: 

44.4% lose one

44.4% wash

11.1% profit one

E.V. -0.33 CP

 

Spend 2 CP:

29.6% lose both

44.4% lose one

22.2% wash

3.7% profit one

E.V. -1 CP

 

Spend 3 CP:

19.8% lose three

39.5% lose two

29.6% lose one

9.9% wash

1.2% profit one

E.V. -1.67 CP

[BrotherAetherick]

Great breakdown. Really shows how much better Grand Strategist and Adept of the codex is compared to the DW and DA warlord traits.

I just hate not being able to use the blood angel datacards for maelstrom when using a soup warlord.