Category Archives: Probability

Retrospective

With 2012 now done and dusted, I’m taking the opportunity to highlight some of the greatest hits of WarHamSandwich.  In particular I’m homing in on the articles that have retained their relevance even with the many changes from 5th ed to 6th ed.

So, happy new year my friends, and check out some of the greatest hits below:

2013 will see plenty of new topics, and of course the updating of some old favourites for 6th edition.

Best wishes and a Happy New Year to you all!

Merry Fleetmas

I already gave you some basic rules for how to pick which dice to reroll for fleet, but they are specifically for charging in the open.  Charging through terrain means 3d6 drop the highest, which means more complexity in terms of which dice to reroll.

For your benefit dear reader, I’ve gone ahead and worked out the optimal rerolls, but they do require a more clunky set of guidelines than last time.  A key distinction here is that when you make your initial 3d6 roll I’ve called the lowest die the min, the highest the max, and the middle one the mid – the best choice for a reroll depends on what you got for min mid and max in your first roll.

Without further ado, here’s what you need to get the most out of Fleet in terrain:

 Range Reroll all 3d6 Reroll lowest 2d6 Reroll Lowest D6 3″ 111 Not 111 Never 4″ 111 or 112 All others 122 5″ max<=2 All others 123, 133, 223 6″ All others max>=4 mid=max=3 7″ max<=3 All others max=4, mid=3 OR 4 8″ max<=3 All others mid=max=4 9″ max<=3 All others max=5, mid=4 OR 5 10″ max<=4 All others mid+max>=10 11″ max<=4 All others max=6, mid=5 OR 6 12″ No 6s One 6 Two 6s

This probably seems a bit impenetrable at first, so here’s a few notes on reading it.  The range column is the range to your target, the three other columns are the conditions for a given range that you would reroll 1, 2, or 3 dice.  For example, say you are 3” from your target, if you roll triple 1, then you should reroll all three dice, otherwise reroll the lowest two dice.

As another example, say the range is 10”, if the highest of your dice (i.e. max) is 4 or less then reroll all three, if the sum of the highest and middle dice is greater than or equal to ten then just reroll the lowest 1d6, otherwise just keep the highest die and reroll the lowest 2d6.

Happy fleeting my friends!

Terrain Wreck

So I left you last week with a cliffhanger, namely how much does terrain impact your odds of reaching assault?  More specifically:

Is Barra the Berzerker better off charging guardsmen in the open 8” away, or going for a squad in terrain 6” away? What about with rerolls, or with Fleet?

When terrain comes into play, the assault range mechanic gets a little more complicated.  Instead of a straight 2d6 roll, it becomes 3d6 drop the highest.  To show the difference this makes, here’s a straight up comparison:

So the red line shows the probability of achieving an assault distance in the open, and the green shows the same for terrain.  Unsurprisingly, terrain always makes it harder to reach your opponent, but what’s more important is by how much.  I’ve highlighted two points here as they give us the answer to part of the question I posed.  At 8” the berserker has a 42% chance of reaching the guardsmen in the open, but at 6” he has a 48% chance of reaching the guard in terrain.  The guardsmen in terrain are the right choice of target.

But what about rerolls for the Icon of Wrath?  How does Fleet affect it?

The chart may seem daunting at first, but hopefully a quick bit of explanation will make it more accessible.  The three red lines are the odds of assaulting a certain distance in the open with no reroll, with a reroll, and with Fleet.  The three green lines are the same but for assaults through terrain.  To address our specific question about 8” in the open vs 6” in cover I’ve again added the relevant dots with percentages.  These clearly show that in a ‘like for like’ comparison (i.e. with a reroll, with fleet) it’s always better to assault 6” in terrain than 8” in the open.  So to answer the original question, Barra the Berzerker should go for the guardsmen in cover in all of our scenarios.

That said, I didn’t build the chart just for this one scenario, you can use it to compare all kinds of situations – and can draw some interesting insights from examining it.

For example, comparing the darkest red line with the darkest green line, we can see that a unit with Fleet charging through terrain has equal or better odds of making assault than a normal unit charging the same distance in the open.

Similarly, checking out the mid-tone green line versus the dark red shows us that when assaulting though terrain units that can reroll charge distance (not Fleet) are also better than normal units charging in the open but only for distances up to 6”.  After this point we see a crossover in the lines and a normal charging in the open is better from 7” onward.

Even simply taking a look at the normal line, vs the reroll line, vs the Fleet line can really highlight how big a difference these abilities can make.

I do have to apply the caveat that my analysis assumes that the player using Fleet always makes the optimal choice for which dice to reroll.  But you’ll have to wait for those rules until next week…

A Pinch of Assault

Assault ranges were one of the biggest changes in the move to 6th edition.  In total opposition to 5th ed, we gained the right to premeasure the charge range, but lost the ability to know exactly how far we could charge.  I’m sure Heisenberg would be proud… So here’s a question for you:

Barra the Berzerker Champion is leading his unit to collect more skulls for the brass throne of mighty Khorne.  8” to his left in open terrain is a squad guardsmen, 6” to his right is another guard squad in cover – which is he more likely to reach?  If he had a banner of wrath how would that affect his chances, and what if it had been a unit of Possessed rather than of Berzerkers?

For most units in 6th ed, the charge range is determined by the sum of 2d6, or if in terrain, 3d6 dropping the highest.  The mechanics get more complicated as some units can get a reroll, and others can reroll specific dice (i.e. Fleet).  As most of you know, 7 is the average result on 2d6, but this can lead to the unfortunate assumption that you should expect at least a 7.  I’m sure as more people get some games in and launch assaults in 6th, they realise that this is not the case…

Let’s look at the no terrain mechanic first.  I’ve charted the odds of making the charge for a straight 2d6, 2d6 with a reroll, and Fleet (i.e. reroll either or both dice).

Unsurprisingly, having a reroll improves your chances, and being able to pick and choose which die to reroll increases your odds even further.  That assumes you know when is the best time to reroll just one dice (i.e. my results assume you make the optimal choice in all instances).

We can see from the chart that at 8″ our Berzerker friend has only a 42% chance of reaching the guardsmen in open terrain.  If his squad has the Icon of Wrath, then he can reroll charge distances, which would bump his odds of reaching them to 66%.  A Possessed Chaos Marine has Fleet, which enables him to pick and choose which dice to reroll – raising the odds of reaching those guardsmen to 72%.

To make life easier for you dear reader, for units with fleet, I’ve got some rules to make sure you make the best reroll you can:

Range: 3” or 4” reroll both

Range: 5” reroll anything <3

Range: 6” to 10” reroll anything <4

Range: 11” reroll anything <5

Range: 12” reroll anything <6

Note1: when I say reroll anything <x I mean if both dice are <x reroll both, and if one die is <x reroll just that one.

Note2: If the above rules don’t work for a specific result then reroll the lowest die only e.g. range 10” and you roll a 4 and a 5 (say), just reroll the 4.  In the case of doubles, just arbitrarily pick one for the reroll and keep the other.

So what about the terrain?  In the question above the Guardsmen in terrain are closer, but does the terrain make it less likely to reach?  You’ll have to tune in next week for that one my friends!

Krak vs Frag

One ‘fast call’ that often crops up is the decision of Krak vs Frag missiles when targeting MEQs. So here’s a question:

Lothar the Longfang stands atop a bastion with clear sight to a squad of chaos marines. He sees one in the open, with four comrades in cover. Should he use a krak on the one, or frag on the four? What if all 5 were in cover?

6th edition has introduced the concept of focus fire, whereby you can pick out individual models in the open, and has also reduced some of the standard sources of cover save (e.g. area terrain, and firing through enemy units). Both these changes affect the decision to go frag or krak vs armored targets.

Let’s take the simple stuff first, say Lothar fires a krak at the marine in the open – how does that pan out. With 3’s to hit, and 2’s to wound theres a 56% chance of killing the marine. What about the frag on the four? Well the scatter die is generally worse than all but the lowest BS values, so Lothar’s odds of hitting will likely drop (though this is quite dependent on the layout of the target models and any other nearby models that could be hit by a scatter). My previous analysis, puts BS4 scatter as about equivalent to BS3, so lets say 4+ to hit, 4+ to wound, and 3+ armor saves; and let’s say one blast marker can cover all four targets. This gets us an expected kill rate of less than 37% – worse than the krak. In fact, you’d need to cover about 7 marines before it was as effective as 1 krak shot (with no cover).

Of course, opponents are not always so obliging, and all your targets may be in cover. With 5+ cover that Krak missile drops to a 37% chance of a kill, and 4+ cover drops it to 28%. So how does the frag compare in these circumstances? A frag blast that covers 3 marines is roughly equivalent to 1 krak into 4+ cover. In 5+ cover you need about 5 marines covered to be worth your while.

So what’s the answer to our question? Well, with some dope in the open, Lothar should krak away rather than frag the four. But when it’s five in (say 5+) cover then it edges towards frag – but there’s something you should keep in mind:

They are roughly equivalent but there’s a key difference here – I’m using the ‘expected’ number of wounds to compare, but a krak can only kill 1 marine whereas the frag can kill multiple. The catch is: while the frag can kill more, your are also more likely to walk away with no kills at all. The concept here is volatility – if you need that one kill then the krak is more reliable even in situations where the average says they are equivalent. Sometimes though, you need to take the chance on getting multiple kills (e.g. to force a break test) and for that you need the frag.

Fast Calls

In the heat of battle you have to make all kinds of on the spot decisions. Sometimes these are judgment calls with no analytically right or wrong answer – you just go with what feels right. However, sometimes even very good players do this in situations where a bit of thought would lead them to a definitively right choice.

For example, during the 2012 ETC, my opponent and I were discussing a cover save for a squad of my chaos marines. I contended that they were in 4+ cover, he said no. After some debate he offered me to ‘4+ it’ to determine if they were in cover, to which I offered a flat 5+ cover save which he accepted. In a later game the same situation arose with a squad of Obliterators, this time I agreed to ‘4+ it’. So here’s a simple question for you:

Was I right to take a different approach to two seemingly identical situations?

In the case of marines, they needed cover to have any save at all, rolling a 4+ to see if they are in cover of 4+ needs two 4+s which is only 25% likely. Arguing for a 5+ got me a 33% survival chance. So heres a rule of thumb, if your opponent offers a 4+ on 4+, offer a flat 5+. If an opponent asks for a 5+ get him to 4+ it instead.

Then why the change for the Obliterators? Well, like all good rules of thumb, you need to know your exceptions. The key difference is that if the Obliterators aren’t in cover, they still have their 5+ invulnerable – arguing 5+ cover would achieve nothing. I should roll for the 4+ to try for 4+ cover and if I’m not in cover then I can still use the invulnerable.

There was a similar situation against Imperial Guard. The IG order “Fire on my target” forces you to reroll successful cover saves. A quick bit of math and I chose the 5+ invulnerable vs 4+ cover, for the same statistical reasoning as before – a 33% chance of survival vs a 25% chance.

Whenever you find yourself in these kinds of situation, take a moment to think about it – because eking out a few extra percentage points can win you a tight game, and aren’t those the best victories?

Basic Anti Air

I realize that it’s been a while since I’ve thrown some numbers at you, and isn’t that really the purpose of this blog? So back on form, here’s a question:

Icarus was the boy of legend who flew too close to the sun, but if he traded his wax wings for a vendetta, then how worried should he be about the skyfire interceptor lascannon that they named after him?

Flyers are the new hotness of 6th edition and everyone needs to have a way of dealing with them in order to stay competitive. The BRB gives us two generic options, the Icarus Lascannon, and the Quad Gun.

Both have the very interesting interceptor rule, which lets them take a pot shot at any enemy unit that has arrived from reserve that turn – so you get a chance to blow that flyer out of the sky at the end of his movement phase (before he gets a chance to fire).

Both fire at full BS vs. flyers (thanks to the skyfire rule), and given that flyers are usually medium armour at best, you’d be hopeful of getting a lucky shot in to pop your opponent’s beloved flying machine of death. Unfortunately the numbers aren’t really on your side

So that Vendetta flies onto the board full of hopes, dreams, and heavy weapons, how does he fare? Let’s take the Icarus Lascannon first.

Assuming BS4, with fours to pen his AV12, and fives on the damage chart to explode him we get an 11% chance of success (which drops to 7% if your opponent evades). The hull point view isn’t great either, we can do at most 1 hull point, but the expected value is 0.3 hull points.

The quad gun is certainly a better bet for doing hull point damage (not so hot for an explodes result). We can potentially do 4 Hull points of damage to that Vendetta! But what are the odds?

Hull Points inflicted vs Odds
0 41%
1 41%
2 15%
3 2%
4 <1%

As you can see, not great. The expected value is 0.79 Hull Points, which is a lot better than the Icarus, but still not a major threat.

Running a few more numbers, I came across a sad truth. A snap firing twin linked tesla destructor (i.e without skyfire) is as dangerous to a vendetta as the dedicated AA Quad Gun.

So to answer my original question, our vendetta pilot shouldn’t sweat it, but my overall conclusion can be summed up as:

God damn necrons.

Vindication for the Vindicare

A long long time ago, a gentleman called ‘JohnnyCache’ asked:

“I would like to see a breakdown of my likelihood of stopping a vehicle with the 4d6 turbo-penetrator round from the vindicare covered in your excellent mathblog, because I am looking at it and it looks like (assuming he hits the vehicle) he will glance-or-better armor value 10 9 time out of ten? And glance-or-better av 14 over half the time?”

I sent him a quick response without writing an article, and didn’t think anymore about it until today, where I realised that I had completely misinterpreted the vindicare’s tank killing capability.  So firstly apologies to JohnnyCache, but better late than never, right?

I’ll frame the discussion with one of my more traditional questions:

How many marines with lascannons would you need to equal the Land Raider killing potential of one vindicare turbo penetrator shot?

The Vindicare has long been one of my favourite models both in terms of looks and rules.  What a badass.  In the current ruleset the Vindicare is BS8, and has a special ammo type called ‘Turbo Penetrator’ that has 4d6 for armour penetration.  When I responded to JohnnyCache, I basically provided the following:

Which basically proved his maths correct, 55% chance of glancing or better on AV14, 90% chance of glance or better on AV10 (assuming you hit).  But it was only today that a few additional details popped into my head.  Number 1, his weapons are marked as ‘Sniper‘, so they get a Strength of 3 to add to that 4d6, giving us the following:

The AV14 penetration result rockets up to over 75%!  But wait, there’s more: ‘Sniper’ weapons are also rending, so any roll of a 6 on penetration gets an additional d3 to penetration – combined with the Str 3 bump we get the following:

As you can see these combine to form a major boost to the previously predicted tank busting power.

Looking back to my setup question, the Vindicare is BS8 (approx 91% chance to hit), penetrates AV14 79% of the time, and the weapon is AP1 so it adds +1 to the damage roll.  End-to-end odds of killing a Land Raider? approx 37%

How many devastators with lascannons does that equate to? Well with BS4, S9, and AP2 you would need 12 lascannon-toting marines to get the same odds of a kill.

In my opinion, for 145 points, he’s a steal.

A tussle with Tesla

The recent reboot of necrons has injected some new ideas into the 40k meta, and players are trying out various combos and builds from the new codex. The new rules introduce some new mechanics, and regular reader Rowan Sheridan has asked for some insight into Tesla weapons.

Let’s kick things off with a question (actually TWO questions):

A nasty necron immortal is firing his Tesla Carbine at a horde of terrifying Tyranids. As he aims, he gets blasted by Paroxysm making him BS1, how does this affect his odds of getting the bonus Tesla hits?
What if his carbine was twin linked due to the Targeting Relay of a Triarch Stalker?

A Tesla weapon gets you three hits on a to hit roll of a six, and so we can quickly get an answer for the first element of our discussion. The BS of the firer of a Tesla Carbine is irrelevant for the purposes of the bonus hits, you always have a one in six (16.67%) chance of rolling a six. Obviously with higher BS you’ll miss less and will therefore always come out better on average in terms of regular hits (just not in terms of bonus hits).

So first portion of the discussion done, bonus hits are a straight 6 on a dice roll, so your BS is irrelevant and your odds are 16.67%. But, what about the second part of the question – Twin Linked Tesla? This is where it gets interesting…

The twin linked rule only allows you to reroll misses, and low BS means more misses, and therefore you are more likely to get a second chance at rolling a 6. The unmolested necron has a BS of 4, has a 16.67% chance of getting a six first time, and a 33% chance of getting a reroll (i.e. missing the first shot). The necron suffering from Paroxysm has a 16.67% chance of getting a six first time, and an 83.33% chance of getting a reroll (i.e. missing the first shot). Lets take a closer look at those odds:

• BS4 chances of getting bonus hits: 16.67%
• BS1 chances of getting bonus hits:16.67%
• BS4 Twin Linked chances of getting bonus hits: 22.22%
• BS1 Twin Linked chances of getting bonus hits: 30.56%

That’s right, a paroxysm affected necron has a significantly improved chance of getting Tesla bonus hits when his weapon is twin linked. In fact the odds smoothly reduce as BS goes up (for twin linked weapons):

• BS1: 30.56%
• BS2: 27.78%
• BS3: 25%
• BS4: 22.22%
• BS5: 19.44%

Now as interesting as this may be, we can’t ignore the fact that at BS1 he’s simply going to miss more often – is the additional probability of the Tesla bonus good enough to mitigate his lousy aim? How about a graph of the total number of hits (regular + bonus) we would expect (on average) per hit roll made?

The graph shows the expected number of hits for BS values from 1 to 5 for a ‘Basic’ (i.e. non Twin Linked, non Tesla) weapon, a Tesla weapon, and a Twin Linked (TL) Tesla weapon.

As you can see for increasing BS you expect more hits, so the Tesla game mechanic hasn’t done anything too strange. In non twin linked Tesla the bonus improves all BS equally, and for twin linked it improves weak BS more noticeably.

If GW had set the Tesla bonus hits at higher level then you really would start to get strange effects. Non-twin linked Tesla would almost negate any difference between BS values, and for Twin linked Tesla low BS could be for better than high! Thankfully that’s not the case.

That’s all a bit theoretical, but Rowan’s actual question was more specific to the Tesla Destructor, which is four twin linked shots of tesla.

Because of the Tesla effect, these 4 shots can get from 0 to 12 hits, but the odds are very much unlike any other 40k mechanic. To show what I’m talking about, here’s a graph of the results of 4 Tesla shots at BS4, Twin Linked and Non-Twin Linked:

If you recall my article on 2d6 we saw a spike on the 7 result because the largest number of possible combinations added up to 7. We see a similar but more complex action here where certain results peak as there are more ways for that exact number of hits to occur (as an interesting quirk, it’s impossible for the 4 shots to produce a result of 11 hits). The key difference between the TL and non TL results is that TL gets better odds of the bonus hits and so the combinations that need those ‘3s’ get amplified (and the improved hit rate that makes TL generally useful also shifts the curve to the right).

Interesting graph, but how about I boil it down to a straight comparison?

• 4 Tesla shots at BS4 gets an average of 4 hits
• 4 Twin linked Tesla shots at BS4 gets an average of 5.33 hits

And if you think that’s good – I haven’t included Arcing in this analysis!

Psychic Defenses (Part II)

Please Note: I’m moving country at the moment so the ‘polish’ on my articles for the next couple of weeks may be a bit lacking.  The content will still be relevant, but just not packaged as well as usual.

I recently wrote up an analysis of psychic defenses, breaking them into two types: one that tries to block the power after it has passed the psychic test, and a second that simply tries to make it harder to pass the psychic test to begin with.

It grew some legs on DakkaDakka, and a few good readers pointed out that it’s hard to use my work for direct comparisons as the odds of initially passing the psychic test wasn’t factored in.  This was actually intentional (and noted in the post itself) as the odds of passing created another variable to track (namely the leadership of the caster).  But given the feedback I thought I should revisit the analysis to provide something more user friendly.

So here it is.

I’ve done an end to end calculation for caster leaderships 10, 9 and 8 showing the total odds of passing the psychic test and bypassing the defense as one percentage.  By necessity it’s a separate table for each of the three leadership values, with each row representing some form of particular defense (or in the case of runes of witnessing, a bonus).

So the left column is the particular situation, and the right is the odds of the power being successful.

So with a caster Ld10, ye olde Psychic Hood is not brilliant since the defender can at best match the caster Ld.  So it lags behind Runes of Warding, Rune Priests, and Reinforced Aegis.  (I will note here that I have not factored in the differences in how and when the psychic defenses can be used, e.g. the range difference between Runes of Warding and Shadow in the Warp is huge but I’m focusing only on the dice mechanics here so I leave it to the reader to know and interpret the context accordingly).

At caster Ld9 the Psychic Hood starts to get a look in.  If the defender has Ld10 to the caster’s Ld9 then it gives a better defense than Rune Priest, Runes of Warding and Shadow in the Warp.  Reinforced Aegis still beats it on the odds though.

At caster Ld8 the Hood gets better again.  With an Ld10 defender the odds of a successful psychic power is only 20%.  Again this beats Rune Priest, Runes of Warding and Shadow in the Warp, but Reinforced Aegis still maintains better odds.