Calculating the Odds: Effective Health and Legion

     Earlier this year a different game I dabble with was having a debate.  There was an option that gave a unit a second armor save on a 5+.  The common response was this meant it had 33% more health.  A better response was 50%.  These are both incorrect.  To hit that conclusion took a bit of math, and I was also curious how to apply these calculations to Legion.

  [Math Section Begins]

    Effective health is essentially the health of the model taking into account its armor save.  Armor saves are a quasi-infinite chain.  As long as you keep making them, you don't lose health.  You can model this as a series of events:

This is a geometric series.  It can be expressed more succinctly as:

Thankfully, as the odds of an armor save are always lower than 1 it also converges!  This means that it nicely collapses down to this:

Finally,  I was curious about Dodges here too:

The real wrinkle with dodges is you do have a chance to be overkilled.  This effectively makes the dodge worthless but is difficult to model.  It is a good reminder that dodges on a near dead unit are of less value though!

[Math Ends]

What does this mean for Legion?  First, Effective Health is how many hits you need to score on a unit to finish it.  Secondly, two Charts!

Legion Die	40k Armor Save	Odds	Effective Health
White Die	6+	0.1666666667	1.2
White Die + Surge	5+	0.3333333333	1.5
Red Die	4+	0.5	2
Red Die Plus Surge	3+	0.6666666667	3
	2+	0.8333333333	6

	Health Per Model	Effective Health / wound	Effective Health	Unit Health(3)	Unit Health(4)
Wooks	3	1.2	3.6	10.8	14.4
Wooks(surge)	3	1.5	4.5	13.5	18
Magna	2	2	4	12	16
Magna(Surge)	2	3	6	18	24
IRG	2	2	4	12	16

    Is it any wonder that a unit of Magna are harder to kill than a unit of Wookies?  What really jumps out at me is how much more a red die save benefits from surge on the defense than white.  Looking at the delta, a full unit of Wookies with plentiful surges gains 3.6 health.  A full unit of Magna guard with the same surges gains 8!

What about the dodge results?  How does effective health effect the value of a dodge?

White Die	1
Red Die	1
Red Die Surge	1

This ... is annoying but not entirely surprising.  Leaving it to the reader, from the formula for both effective health and dodges a few simple steps will show that this does collapse to 1.

    Overall this leaves me with a few takeaways.  It strikes me that people point towards the free dodges the Magna get as to why they are so tanky.  However, I feel that I've been sleeping on the roll that surges play here.  If you can maintain surges on a unit of Magnas its worth effectively 8 dodge tokens.

    The second takeaway is Into the Fray is something to experiment with more.  Currently if I give Magna/IRG/Dews an upgrade it tends to be tenacity.  The above results indicate that maybe a mistake, after all only living units get to generate attack dice!

    From the above I've been experimenting with the best way to generate these surges in Empire.  Building an Empire list that helps Dews or IRG stick around as long as Magnas should be a thing.  Hopefully I'll be able to get some table time with that soon and see if Theory and Practice are the same.