Category Archives: Probability

21 September 2011

Psychic defenses

By WarHamSandwich

This time it’s a request from the gaming club The Warheads and we are going to cover Psychic powers, or more specifically things that stop them.  So let’s set up a few scenarios.

A sinister Slaaneshi Daemon Prince tries to cast Lash of Submission on a squad of Space Wolves near a Rune Priest, if the Prince tried the same on a squad of Grey Knights near a Dreadnought with Reinforced Aegis, would he be more or less likely to succeed?

There are a wide variety of psychic defenses out there, and they use different game mechanics to achieve the same objective i.e. to stop the psychic power.  To successfully use a power the psyker must roll less than or equal to his leadership on 2d6.  Most psykers have a high Ld and this isn’t something you’d generally expect to fail very often.  The mechanics used to stop them fall into two types:

Type 1) effects that make it harder to pass the psychic test e.g. reducing the Ld value of the Psyker (Aegis, Reinforced Aegis) or increasing the number of dice rolled for the test (Runes of Warding, Shadow of the Warp)

Type 2) effects that happen after the psychic test to try to block the power, e.g. the leadership roll-off for a psychic hood, or the 4+ nullify from a Rune priest.

Spacewolves can have a special ‘layered approach’ where they get a 5+ nullify from a Wolf Tail Talisman, and if that fails they can then try the 4+ from the Rune Priest.  If the Rune Priest happens to be Njal Stormcaller then that 4+ becomes a 3+.  But even with all that wolf-cheese, the master of psychic defense has to be Khorne.

The ‘Blessing of the Blood God’ gives a 2+ invulnerable save against any wounds from psychic powers, and his most beloved champion Kharn is flat-out immune to psychic powers.  Though to be fair these abilities are not widespread, and the Chaos Marine codex has pretty much no psychic defense.  (Update: it’s been pointed out by a few people that Lady Malys of the Dark Eldar is also immune to psychic powers, as is any unit she joins – so it’s not just Khorne who can totally nullify psykers!)

The effectiveness of Psychic Hoods varies depending on the relative leadership values of the defender versus the caster.  If the caster has a low Ld compared to the defender then he’ll have a very tough time.

So for all these ‘Type 2’ defenses I’ve simply calculated the odds of blocking the psychic power (assuming that the psychic test is passed).  For the hood I’ve calculated it for all possible differences of leadership value between caster and defender.  I’m sure the range encountered in practice is smaller, but I did it simply for completeness.

The table is ordered from best defense to worst defense as follows:

Psychic defense Chance of Nullify
Psychic Hood with Caster Ld 6+ less than Defender 100%
Psychic Hood with Caster Ld 5 less than Defender 97%
Psychic Hood with Caster Ld 4 less than Defender 92%
Psychic Hood with Caster Ld 3 less than Defender 83%
Blessing of the Blood God 83%
Wolf Tail Talisman AND Njal (5+, 3+) 78%
Psychic Hood with Caster Ld 2 less than Defender 72%
Njal  (Straight 3+) 67%
Wolf Tail Talisman AND Rune Priest (5+, 4+) 67%
Psychic Hood with Caster Ld 1 less than Defender 58%
Rune Priest (Straight 4+) 50%
Psychic Hood with Caster Ld = Defender 42%
Wolf Tail Talisman  (Straight 5+) 33%
Psychic Hood with Caster Ld 1 more than Defender 28%
Psychic Hood with Caster Ld 2 more than Defender 17%
Psychic Hood with Caster Ld 3 more than Defender 8%
Psychic Hood with Caster Ld 4 more than Defender 3%
Psychic Hood with Caster Ld 5+ more than Defender 0%

Ok so that’s Type 2 defenses, what about those Type 1s?

I start by graphing how the odds of passing the psychic test vary with increasing leadership, for a variety of scenarios (including Runes of Witnessing which are not a defense at all, they are there to improve a casters chances of passing).

So we get a set of S curves, where the red curve represents a ‘normal psychic test’ and any curve below it is hampered by a defense.  The lower the curve in the graph, the stronger the defense.  As you can see the Reinforced Aegis is the meanest of the lot.  It is worth noting that the Runes of Warding give a perils result on any result of 12 or above, and increases the odds of a double 1 to push the odds of perils from about 5.6% up to nearly 45%, that is a serious threat in its own right.

While this clearly shows the effect of the defenses, it can be a bit tricky to read the actual impact of the defense.  So with that in mind I’ve also plotted the difference each defense makes for a given Ld value, shown below.

This shows specifically what these defenses are worth against a given Ld.  The Normal Psychic test is a flat red line as it compares exactly to itself (if you get what I mean).  Runes of Witnessing are above the line as they improve the odds of casting, and the defenses are below the line because they reduce the odds of passing.

So what about that original question?  Which target is easier for the Daemon Prince.  Well he’s Ld10 so under normal circumstances he has a 92% chance of passing his psychic test, the Rune priest has a 50% chance of blocking him if he’s successful giving an overall probability of 46% for making the Lash.  With the Reinforced Aegis, the odds of making the psychic test drop to 42% making it slightly more effective than the Rune Priest.

I’m sure there are lots of particular scenarios you could come up with, and hopefully I’ve given you the data to be able to make like for like comparisons, but don’t be afraid to put a request in for a specific situation if you can’t work it out!

1 Comment | posted in Probability

14 September 2011

When Lance met Melta

By WarHamSandwich

It’s always great to get requests from readers, as it makes me consider scenarios that I don’t always encounter in my own games.  This week’s request comes again from the good folks of the Penny-Arcade forums, namely OtakuD00D, and Mr_Rose

“I want to see the odds involved in Darklight Lances, can you include Void Lances too, please? Curious to see what Heat Lances can do too”

Right, so Dark Eldar have been given a pretty amazing facelift of late, and come with a few new twists in terms of weaponry.  They have plenty of lance weapons (which count armour over 12 as just 12) and also have access to melta weapons – in particular a melta lance weapon!

So, as I said, lance weapons count armour over 12 as 12, but another way of looking at this is to say that a penetration result of 12 will glance any armour, and a penetration result of 13+ will pen any armour.  So for our comparison we can treat a 13 result as being the same as a 14 or 15 (or 1,000,000 for that matter).

I’ve plotted this using my usual format, and I’ve included Lances with Str 8 (Dark/Bright lance), S9 (Void Lance), and S10 (Orbital Lance Strike).  As a baseline I’ve put in Krak missiles, and a Lascannon.  Last, but by no means least, I’ve also included the S6 Lance Melta combo (aka Heat Lance).

So lance weapons have a peculiar shape in that once they hit 13 then they basically flatline keeping the same odds ad infinitum.  As I noted above this is because all armour over 12 is just 12 to a lance, so once you hit 13 you’ve got the pen on anything.  In terms of comparison with Krak and Lascannon, you can see that the S8 lance follows the Krak until they hit 13 and then the krak gets progressively worse while the lance keeps the same value (similarly for the S9 lance vs the lascannon).

A more interesting comparison is the S8 lance vs the lascannon; the lance is consistently worse until we’re looking to penetrate AV13.  Reading the chart you can see the lascannon has a 33% chance to pen AV14, the same as the S8 Lance (there’s a bit of a flaw in how I represent the lances in the chart, as once you reach 13 they are always penetrating whereas a 13 from a lascannon would only glance AV13.  Rest assured I’ll find a better way of presenting this type of data in the future).  For AV14 the S8 lance stays at 33%, while the lascannon sinks to 17%.

So to sum it up, lances only get interesting against AV13 and AV14.  The S10 Lance is particularly lethal against AV13/14 witha  67% chance of pen, the S6 Melta (at close range) is also pretty savage with 58% chance of pen, which is about the same as a standard S8 meltagun against AV14.  Next in line is S9 Lance, and then S8 Lance which is as good against AV14 as a regular S10 weapon.

I’m focusing on AV14 here, but don’t be misled – that’s the ‘best case’ scenario for lance weapons.  The higher the AV above 12 the more benefit the lance brings.  So while a S8 lance is as good as S10 against AV14, S10 is better against AV13 and below (significantly so for AV12 and less).  This effect is particularly well illustrated by a comparison of a S8 meltagun, a S6 meltagun, and a S6 lance meltagun.

The benefit of the lance is clearly shown when the green line goes horizontal, but you can see that that it only catches up with the regular meltagun (red line) at AV14, it’s noticeably worse at everything else (and won’t instant death T4 models!).

Ok so, summary.  Lance weapons are only beneficial against AV13 and 14, moreso for AV14.  Any (non lance) weapon that equals their effectiveness at AV14 will be better than them at everything below.  Weapons that are weaker against AV14 could well be superior against lower AVs (e.g. a lascannon is worse than a brightlance at AV14, is equal at AV13, and better at everything below).

Final verdict – it’s very meta sensitive.  In a world of razorback spam, lance weapons do you no particular favours.  But if your usual group is running triple Land Raiders then you’re golden.

2 Comments | posted in Probability

31 August 2011

More penetration: Ass versus Las

By WarHamSandwich

This weeks sandwich request comes from the inimitable Paul ‘Mandragoran’ Quigley (just back from Switzerland as part of the Irish team in the 40k world championships!). We’re taking another look at penetration today, this time factoring in the impact of rending. A useful reference point would be here where I’ve covered the efficacy of a variety of anti tank weapons, but the material in this post does stand on its own too. Right, question time:

A Black Legion tank commander is navigating his Land Raider through enemy territory. He’s caught in crossfire between two Razorbacks, one left, and one right. Both are at the limit of their weapon range, so the commander decides to rush towards one so that he can’t be hit by both. Should he drive toward the Assault Cannon armed Razorback, or the Lascannon armed Razorback?

So, rending against vehicles. You get your usual 1d6 plus weapon strength roll, with the added twist that if you get a 6 on the roll, then you get to roll a d3 and add it to your result. This means that an assault cannon at Str 6 could get a 15 and penetrate a Land Raider (Str6+roll a 6 on d6+ roll a 3 on d3 =15). The question is, can it do a better job than a Lascannon (which is a plain Str 9 + 1d6 with no rending)?

It’s a little tricky to do a straight like-for-like comparison here as a key feature of the assault cannon is that it gets four shots, whereas the Lascannon only gets one. So here’s my starting point. I’m assuming that there are no misses, and for the Lascannon I’m looking at the odds of a penetrating hit, and for the assault cannon I’m looking at the odds of at least one penetrating hit. Some of you may object to that approach, but bear with me for now.

(Just to note: the gap in the line isn’t a mistake, it’s simply that you can’t get a 12 on the assault cannon as rolling the 6 gets you an extra d3 making you jump from 11 to 13+).

So what have we got: one hit with a lascannon has a (just under) 17% chance of getting a pen on AV14, but 4 hits from an assault cannon gets you (just over) 17% chance of getting one or more penetrating hits on AV14. Okay the odds are only a tiny bit higher but you can get from one to four pens so the net effect can be a lot stronger. So the word on the street is correct, assault cannons are straight up better than lascannons at penetrating AV14?

Well, not so fast.

Let’s go back to those assumptions from earlier. None of the shots miss. “So what?” you say, “the assumption was the same for both!“. Actually it’s different. The odds of getting all hits on a one shot weapon are better than the odds of getting all hits on a four shot weapon (assuming equal BS). The analysis above assumes that the Razorbacks never miss, so the more unreliable the firer, the less accurate that graph becomes.

To illustrate the point I’ve run the same analysis showing the results for 4, 3, 2 and 1 hits on the assault cannon versus the lascannon.

The comparison is no longer quite so clear cut. We need to account for the end to end process from hitting through to penetrating. So lets’ do that. Lets assume BS4 for the assault cannon and the lascannon.

For the assault cannon we need:

  • 3+ to hit
  • 6+ to rend
  • 5+ to penetrate

This gives us a 4% probability of success. But we get 4 shots, if you’re thinking that 4 shots at 4% gets you a 16% probability of success then you probably need to read my blog more often; if you’re thinking the answer is 14% then you probably don’t need me at all. (Success here means one or more penetrate results)

For the Lascannon we need:

  • 3+ to hit
  • 6+ to penetrate

This gives us an 11% probability of success, and since we only get one shot, that’s the total odds.

So what’s the final verdict? The analysis clearly shows that neither the assault cannn or lascannon are particularly good at killing Land Raiders, but if they both have the same BS, the assault cannon is definitively better. It is worth noting that the lascannon can sneak ahead if fired by a superior marksman, so a BS5 lascannon is equal to a BS4 assault cannon, and a twin linked BS4 lascannon is better than a BS4 assault cannon (vs AV14).

I say final verdict, but there’s still a little more gas in the tank. I’ve plotted a couple of different weapons so you can check out the relative merits of weapons that I haven’t covered previously. Note that I’ve not done the full end to end calculation here, I’ve simply assumed all shots hit for this chart.

One final weirdness I wasn’t quite expecting, against the humble rhino (AV11) the lascannon is more reliable. It’s basically an artifact of the rend: if you get a 6 then your result ‘jumps up’ out of line with the non rending results. So while we initially were concerned only with AV14, we can in fact make a more general statement: AV12 and above, assault cannon more lethal, AV11 and below Lascannon more deadly!

5 Comments   |  tags: , , , , , , , , | posted in Probability

23 August 2011

Ka-blammo!

By WarHamSandwich

I first touched on scatter dice in my earlier post on deep striking. Blast weapons don’t use Ballistic Skill in the same way as ‘regular’ weapons, but it is still an important factor in hitting your foes. Question time:

A renegade Ordo Hereticus Inquisitor carrying a psyocculum is hunting the battlefield for Mephiston. The inquisitor is joined by his trusty squad of psyker henchmen with the Psychic Barrage (large blast) power. Assuming they pass their psychic test, what are the odds that they will hit Mephiston?

Right, blast weapons use the scatter dice described here. At its most basic, a 33% chance of a hit, and 67% chance of a miss; if you miss then it scatters 2d6 inches but unlike deep striking you can subtract the firing models Ballistic Skill (BS) from the 2d6 result. So if you roll a ‘miss’ but get a distance less than or equal to your BS then that miss becomes a hit (i.e. you don’t scatter). Naturally this means that the higher your BS, the more ‘misses’ get converted into hits, and if it does scatter then it won’t scatter as far.

To show the effect of increasing BS values I’ve pulled together a 3d plot. So each colour represents a BS value, from BS0 at the front to BS10 at the back. The odds of a particular result go from left to right, so taking BS0 as an example, the odds of a HIT is the leftmost blue column (at 33.33%) and the odds of a particular scatter are to the right, e.g. a 2 inch scatter has a probability of 1.85% (for BS0) and a 7 inch scatter would be 11.11% likely (for BS0).

So quick summary on how to read this:

  • the height of a given column is the probability,
  • each colour is a BS value, and the BS values get higher as you go back,
  • the foreground numbers are a HIT (leftmost) or a particular scatter distance (from 1 to 12 inches).

Since the the BS value is subtracted from the scatter distance, you can clearly see the maximum scatter get smaller with each step increase in BS. So looking at the BS10 scatter (all the way at the back in pink) it’s a 94% chance of a hit, 4% chance of a 1 inch scatter, and a 2% chance of a 2 inch scatter.

Given that the the psyocculum gives our Psykers BS10, is that the answer to the question, a 94% chance of hitting Mephiston?

Not quite, theres one more factor to take into consideration. Blast size. The regular blast has a 1.5 inch radius, and the large blast has a 2.5 inch radius. Against a vehicle, only the centrepoint of the blast gets you a full strength hit, but against infantry just clipping the base with the blast template is enough for the full whack.

So in terms of hitting Mephiston, it’s 2d6 scatter minus 10 for BS, and (effectively) minus another 2.5 inches for the radius of the large blast. So we’re subtracting 12.5 from a number that is at most 12, simply put they can’t miss! There aren’t many mechanics in the game that can say that.

The only caveat is that the extra bit of reach from size of the radius doesn’t get you a ‘full’ hit as it doesn’t land exactly where you placed it. You’ll definitely hit the guy you were centred on, but you’ll cover different models around him if it does scatter those one or two inches.

So, back to more general principles. You may recall my uber list of BS rankings, well I’ve now we can add two new charts to that list.

First up, regular BS accuracies with blast accuracies added (in pink). The blast accuracies are the probability of hit but disregarding the radius of the blast (i.e. the odds of the centrepoint hitting your desired target point). Blue columns are ‘regular’ shots, green are twin linked, and pink are blast. Hmmm I guess I left out twin linked blast, guess that’ll have to wait.

Secondly looking specifically at the effect of the radius (no ‘normal’ i.e. non-blast shots on this one) . So these are the odds of hitting with just the centre (in pink), versus blast (in blue), versus large blast (in green).

The pattern is actually pretty simple: BS10 centre is as accurate as BS9 blast, is as accurate as BS8 large blast (and so on down). Essentially each step up in blast size is equivalent to a one point increase in BS.

So there you have it – I thought I had all the BS covered, but there was still more to do; …always more to do.

1 Comment   |  tags: , , , , , , , | posted in Probability

17 August 2011

Going Deep

By WarHamSandwich

Deep striking is a high risk/reward technique that can get your units anywhere on the table, in one fell swoop.  But when things go wrong, they can go very wrong, and on more than one occasion I’ve lost a 225 point unit of Obliterators to a bad scatter.  For that reason I often take some 3 man chaos terminator squads so I only risk 105 points for a chance at a cheeky melta shot.  But how should I be placing them when I deepstrike?  Consider the following:

Chaos Lord Harleck Wynne faces a wall of Imperial Guard tanks.  He has to deepstrike his terminators, as any walking squad or vehicle will be wiped out as it approaches.  Where should his combi-melta armed Chaos Terminators be placed to minimise the risk of mishap? Where should they be  placed to maximise the chances of getting into melta range? Where should they be placed to get a balanced risk of mishap versus melta range?


Ok, so deep striking is governed by scatter dice.  It’s a 6 sided die with two ‘HIT’ faces, and four faces with an arrow.  Place your model where you want him, and roll.  If you get a HIT then you land on target, if you get an arrow, then you scatter 2d6 inches away in the direction indicated by the arrow.  Because the distance is governed by 2d6, the distance follows a pattern already described here such that results of 7″ are the most likely and 2″ and 12″ are the least likely.

The arrows complicate matters as they don’t comply with the discrete probability that I normally use for these calculations, but we’ll touch on that later.

So, with two HIT faces out of 6, we have a 33% chance of landing on target, and a 67% chance of scattering.  If we ignore direction for a moment, then we can take a look at the odds of how far you’ll deviate from your intended location:

That was pretty much as far as my analysis went until quite recently.  This approach clouded my thinking, as I saw it as a straight up question of distance, so I may as well get super close to the enemy as the ‘most likely’ scatter distance was 7″.  Case closed, right?

Wrong!

If you don’t get a HIT, then it’s all about the arrows.  Let’s imagine a model with a 25mm base put on the table in his desired deepstrike position.  He can scatter up to 12″ in any direction, so lets consider a 25″ wide circle as the total space we could end up in (e.g. up to 12″ to left + 1″base + 12″ to the right gives us the 25″, see below).

Time for a fancy graph.  So I plot an area of 25″ by 25″, and represent the probability of landing at a particular point as a height, so we we get a sort of mountainous terrain where the highpoints are where you are likely to land, and the lowpoints are where you are unlikely to land.  In the first instance lets look at the widest case.  So you have a 33.3% chance of landing on target (i.e. a HIT), and a 66.6% chance of scattering.  See below:

As you can see in terms of a single point, the target at the centre is far and away the single most likely final destination.  In fact the difference is so extreme that all you can see of the scatter is some light ‘fuzz’ in a ring around the centre.  So the first point to note is that if you do scatter it would appear that you could end up pretty much anywhere in that 25″ circle we described earlier.  But that’s not particularly enlightening, so lets take a closer look at the ‘fuzz’.

I now remove the HIT from the chart, and the scale can then be changed to show the variation in odds for the scatter results.  It’s worth noting that I didn’t solve this analytically so we don’t get a smooth and pretty set of results, we get a somewhat noisy set of peaks and valleys.  But it’s still good enough to gain some insights and is still essentially representative of how it works in reality.

So as you can see from the dark blue peaks, the most likely area to scatter into is a ring around the target point, (specifically a ring with its edges about 5″ to 9″ away from the target point).  This is an expected result from our knowledge that the scatter follows the same triangular shape of the old 2d6 chart.  Do note how low the odds of landing at any particular point is: about 0.2% to 0.4%, tiny!  Working through the numbers, here’s a simplified version:

So this is a lot of exposition and I haven’t addressed the opening question at all!  What about those terminators?

Based on the calculations above, I carved out the probability of landing in a ‘safe’ area depending on how far away you place the terminators.  But that in isolation is not enough.  We want the terminators to land within 6″ of the tanks to get some hot melta goodness going.  So here I’ve plotted the odds of landing safely for a given distance, and also the odds of ending up safe AND within melta range for a given drop point (i.e. the point you selected to drop at, not where you end up after scattering).  So on this graph the x-axis is the distance from the tank wall you place the model initially, (i.e. before rolling for scatter).

The results weren’t quite what I was expecting going in, though do bear in mind that these findings are only true for the specific set up of the question – this graph isn’t a general rule for all deep strike situations!

So, what does this show?  Well, assuming the parking lot of tanks is the only other unit in the area then unsurprisingly the further away you place them the less likely they are to scatter on to the enemy and mishap.  But playing it safe won’t necessarily get you within the all important 6″ melta range.  Here’s the interesting bit, I had originally thought that putting the terminators 1″ away from the tanks would get you the highest probability of being in melta range with a trade off of slightly higher odds of mishap.  But I was quite wrong.  The odds of getting safely in melta range stay pretty flat if you originally place the model between 1″ and 6″ away, but the odds of a mishap are about 45% at 1″ but fall to about 25% at 6″.  So the tradeoff I mentioned in my opening question, doesn’t really exist – you can play it (relatively) safe and still go for the close range shot.

Lesson learned, drop those terminators about 5 or 6 inches away and you’re playing the right odds.

So how about a more general rule of thumb then?  This specific case aside, how do we make better deepstriking decisions on the fly?  In my opinion, the best general approach is to think in terms of area.  Visualise the 25″ circle around any particular drop point (some assistance here and here), and then look at the friendly and enemy units in that circle.  Now imagine a 1″ buffer around enemy units, and try to estimate what fraction of the circle’s area is covered by all the units and that buffer.  This is key to estimating the risk.

I’ve illustrated a few simple examples below; in each case the centre of the circle is where you initially place the model (i.e. before rolling for scatter), and the red areas have units or other features (such as impassable terrain) that would cause a mishap (don’t forget the 1″ buffer around enemy units!).  Do note I’m assuming that the centre point is a legal placement.  Also note the maths below isn’t quite exact, but is good enough for tabletop guesstimation.

So there you have it – even deep striking right up into someone’s face is not quite as risky as it looks.

Tune in next time when I apply all of this to blast weapons…

3 Comments   |  tags: , , , , , , , , | posted in Probability

12 August 2011

Stacking the Odds Part II

By WarHamSandwich

The previous post on the probabilities for making lots of saves generated a bit of interest, and (as usual) some clever readers pointed out scenarios that should bear further analysis.  Altmann from the Penny Arcade forums asked:

“Can you work in the probability of making 4+ feel no pains as well? I know we’re getting into NASA shit but I’m curious”

Followed by Joe “Maynard” Cullen (of WarHeads fame) who pointed out that some wargear items also add complexities:

“The Wolf Tail Talisman gives a 5+ invulnerable save that happens before the armour save”

So in a similar fashion to my ultimate Ballistic Skill chart, I took it upon myself to rank the performance of a variety of armour types with rerolls, with Feel No Pain (FNP), and just plain regular saves.  This will give some insight into the relative merit of the saving throws we normally encounter in 40k.

As with Stacking the Odds Part I, the chart shows how likely each type of save is to take no casualties from an increasing number of saves.

So chart number one:

This charts the various types of save (and combinations) showing the odds of taking no casualties for up to 6 saves (I cut it off at 6 as about half of them approach zero at this point).  The legend on the right shows the ranking from best to worst with a 2+ rerollable save being the best, and a regular 6+ save being the worst.  The sharper eyed in the audience may notice that some of the save types listed in the legend don’t show up in the graph – namely “5+ FNP” and “3+ FNP”.  Rest assured this isn’t an error, it is simply that they are coincidentally covered by other save types that perform identically.  So a 5+ with Feel No Pain save works out the same as a regular 3+ save, and a 3+ with Feel No Pain save works out the same as a regular 2+ save.

Or do they?

The calculations are correct, but you need to interpret the data in the context if the game itself.  So saving on a 5+ followed by a 4+ for FNP is statistically the same as a 3+, until you get hit by an AP5 or AP4 weapon, at which point all you get is the FNP, which is just a 4+ save (as you can see from the chart is a lot worse than a 3+).  The FNP could also be blocked by a high strength AP- weapon, leaving you with just a 5+.

In a similar vein, the 3+ with FNP is the same as a 2+, but what if they got hit by a battle cannon? the 3+ is negated by AP3, and (assuming we’re talking about T4 units) the FNP is negated by the instant death rule.  So no saves of any kind!  But a squad of terminators would still get their 2+ and (assuming that 5 are wounded by the blast) they have a 40% chance of taking no casualties at all!

So what about those opening questions?  Well Altmann was interested in the effect of FNP on terminators, and to show the difference I’ve scaled the number of saves taken up to 30, and dropped the weaker save types.

The effect is actually pretty strong, if we take say 20 saves, a regular terminator squad has only a 3% chance of being unharmed while the FNP terminators have an 18% chance (again assuming that they aren’t hit by something that negates FNP!)

Joe’s suggestion of the Wolf Tail Talisman (WTT) is charted below. Assuming the squad has power armour, then it works out quite close to (but slightly better than) a 4+ reroll, and worse than a 2+ save.

This should let you compare the various save types available to you, but don’t forget the context of how saves and FNP get negated! If there are any other save types you want to see included, then please do leave a comment.

2 Comments   |  tags: , , , , , , , , , , , , , , , , | posted in Probability

10 August 2011

Stacking the Odds Part I

By WarHamSandwich

This is a topic I touched on briefly before, but I think it’s time for a more comprehensive review.  As I’ve said previously, many players have bad instincts for how odds ‘stack up’, and I often hear comments like ‘if a terminator has to make 6 saves, you’d expect one to fail’.  Time to challenge some assumptions.

A squad of terminators come under bolter fire and have to take 6 saves, what is the probability that they take no casualties? A squad of marines come under a similar hail of bolter fire and also have to take 6 saves, what is the probability that they take no casualties?

At tournaments, and during club play, you may often hear cries of consternation as someone can’t believe that their opponent just made X number of saves in a row.  I think this ties in to certain types of (erroneous) expectations.  Nothing in this game of dice is certain; and you can never get a 100% guarantee of success.

That sounds pretty trite, and taken purely at face value, it is.  But I’m trying to get at something a little deeper.  If a terminator has to make ten saves, or a thousand saves, there is always a chancing of making it, but the odds don’t scale in the way that people expect.  Simply put: a thousand bolter shots is not one thousand times more likely to kill a terminator than one bolter shot; nor is ten bolter shots ten times more likely. As the number of saves to be taken increases, the odds of survival go down in an exponential way (for the non math types that means they start high, and gradually get lower but never quite reach zero).

Here’s one I prepared earlier:

This graph shows how terminator armour and power armour behave as the number of saves to be taken goes up.  Terminator saves are in red, and marine saves are in blue.  The x-axis represents the number of saves that have to be made, and the y-axis shows the odds of the squad taking no casualties for the corresponding number of saves.

So this gives us the answer to the opening question.  If the termies have to take 6 saves, then there’s a 33% chance of them taking no casualties (i.e. find 6 on the x-axis and then look at the corresponding point on the y-axis for the red line).  The other group aren’t so hot, if the marines have to take 6 saves, then there’s a 9% chance of them taking no casualties.

If we extend the analysis a bit, the marines have only a 1% chance of taking no casualties from 12 saves, whereas the terminators have a more respectable 11% chance.  Even at 18 saves the terminators still have a 4% chance of walking away without a scratch.  Now 4% may sound like very long odds, but in truth its not far off the odds of getting a ‘perils of the warp’ result for a psyker.  So not something you’d see a lot, but hardly beyond belief.

One (slightly esoteric) point to note is that this is a ‘memoryless’ system.  This gets a bit subtle, but what I mean is that the current odds aren’t affected by what happened before.  So if a squad of terminators all survived 6 saves last turn and are now facing 6 more saves in this turn, the odds don’t stack to 11% (i.e. for 12 saves), they stay at 33% (for 6 saves).  Whatever happened in the past doesn’t affect your current action.

Now that we’ve covered some specifics, I’ve taken the liberty to graph the behaviour of saves from 2+ to 6+ when having to make up to 6 saves in one block.  Each coloured line represents a corresponding type of save from red for a ‘terminator’ save, blue for power armour, and so on through to the grey line for a 6+ save.

This follows the same pattern as the previous, it just shows more armour types.  But as an interesting illustration, based on the graph check out the following:

A terminator squad taking 6 armour saves has a 33% probability of taking no casualties

A terminator squad taking 6 storm shield saves has a 9% probability of taking no casualties

A terminator squad taking 6 cover saves (4+) has a 2% probability of taking no casualties

A terminator squad taking 6  invulnerable saves (5+) has a 0.14% probability of taking no casualties

Do keep this in mind the next time you fire your hydras at my obliterators!

1 Comment   |  tags: , , , , , , , , , , | posted in Probability

6 August 2011

The Joy of Penetration

By WarHamSandwich

5th edition games often feature a lot of vehicles, and understanding how best to crack open that armour and feast on the goo inside can be a crucial skill for any player.  With that in mind, here’s a question:

After some poor manouevring by your normally brilliant tank commander, your shiny new Leman Russ gets hit by a Lascannon, and by a Battle Cannon.  Which is more likely to penetrate?

The ambush continues, and the Russ is hit by a Demolisher Siege Cannon, and a Multimelta at close range.  Which of these is more likely to penetrate?

For most weapons, armour penetration is relatively straightforward, i.e. weapon strength + D6.  So, for example, a Krak missile can get results from 9 to 14 (not bad against a Rhino, but smacks of desperation against a LandRaider).  However many of the best Anti-Vehicle weapons don’t follow such a simple pattern.

For example, ordnance weapons roll two dice and pick the highest to add to the weapon strength, and melta weapons at half range get to add 2d6 to the weapon strength. This produces results that aren’t as simple, and can have a couple of quirks.

How does this apply to our terrified tank commander?

The interesting thing about the opening questions, is that it depends on what side the tank is hit from.

Let’s take the lascannon versus battle cannon first.  For a rear shot (AV10) the lascannon is more likely to at least glance, but the battle cannon is more likely to penerate.  For a side shot (AV13) the battle cannon is more likely to at least glance, but the lascannon is more likely to penetrate.  From the front (AV14) the Lascannon is the clear winner and is more likely to glance, and more likely to penetrate.  Not that the lascannon is remarkable against AV14 (with <20% chance of penetrating) it’s more that the battlecannon can’t pen Av14 at all!

It wouldn’t be a WarHamSandwich without some charts so let’s take a look at the comparison.  The graph shows the odds of getting at least ‘X’ for an armour penetration roll with each weapon.  So to get the odds of penetrating AV13, we look at the 14 result as this gives us the odds of getting at least 14 (the 13 result would get the odds of at least glancing).

So we can see a crossover from 10 to 11, and from 13 to 14 where the relative efficacy of each weapon against that AV switches.

So what about that second volley of shooting?
Again it depends on the angle. The Demolisher is more likely to penetrate against side and rear, but once we get to the front this flips and the multimelta becomes more likely to penetrate.  The crossover is clearly shown in the chart, below.

So what’s the point Vanessa?  There certainly are some comparisons where you can unequivocally say weapon X is better at anti-vehicle than weapon Y, but often it’s not so black and white.  With a bit of analysis you can pick the best tool for the job at hand.  Here’s a comparison of some of the common anti-vehicle weapons so you can gauge the relative merits against various armour values.

That said, this analysis doesn’t look at the end-to-end process, from hitting to penning to what you get on the damage chart.  I guess that will have to wait for next time…

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6 April 2011

Cut the BS

By WarHamSandwich

My last analysis on rerolls brought up a couple of questions from the audience, with comments mainly focusing on Ballistic Skill.  One in particular about the merits of the Vindicare assassin got me thinking about the how to evaluate shooting ability.  In line with my usual form, here’s a question:

A lowly guardsman stands in the open, he faces on one side, the mighty Eldar Phoenix Lord Fuegan Burning Lance (BS7) and on the other side Krazy Kullen (BS4) the weapon operator on a Chaos Rhino.  Fuegan’s melta weapon is obviously more potent than Kevin’s twin-linked bolter.  But who is more likely to miss?

Crazy Kevin

So my earlier treatise on rerolls only considered ‘normal’ rerolls where whatever you needed to get the first time was the same for the reroll.  Cunning commenter Caolan pointed out that at very high BS the shooter gets a reroll on a miss, but it’s not the same as the initial roll.  So a BS of 6 gets you a hit on a 2+, and  a 6+ reroll if you miss; BS7 gets the same 2+ to hit, with a 5+ reroll on a miss; BS8 is 2+ 4+; BS9 is 2+ 3+; and finally BS10 is 2+ 2+.

Rather than running the numbers in this post, have a look at my response to his comment at the bottom of this post.  But where I took it a step further this time is to calculate the odds to hit for BS1 to BS10 and for BS1 to BS5 with rerolls.  This gives us an interesting ranking of the relative accuracy of the BS values, and the impact of rerolls.  I’ve highlighted the rerolls in green to set them apart from the regular BS values.  The dark blue is the additional accuracy provided by the reroll (so under the dark blue is the basic odds to hit, the dark blue is the reroll, and the whole bar is the total accuracy).

Lo and behold the answer to our starting question appears before our very eyes.  It turns out that Eldar Phoenix Lords at BS7, are only as accurate as Chaos Rhinos with BS4 twin-linked bolters.  They are equally accurate. Who knew that Krazy Kullen was such a marksman?

What really comes through for me is how good BS4 rerolls are.  I’m known to have a fondness for Obliterators, and with twin-linked plasma AND melta at BS4 you can really see why.  More generally we can see that a reroll is better than a 1 point improvement (or more!) in BS for all cases except BS1 – but I think that’s a fairly well known thing already.

That said, you now have the definitive list from BS10 to 1 with rerolls, use it well my friends.

6 Comments   |  tags: , , , , , , , , , , | posted in Probability

30 March 2011

Undo! Undo!

By WarHamSandwich

Sometimes things just don’t go your way.  We’ve all had that situation where you really wish you could do that roll again.  Thanks to wonders such as twin-linking, or lightning claws, sometimes we can.  Today we are talking about re-rolls, and (as is my habit) here’s a question:

A bolter shot hits and wounds a Warlock on a Jetbike, he needs to make a 3+ armour save to survive.  If the farseer had cast Fortune to give him a reroll on that save, how much more likely would he be to survive? and would he have been better off in Terminator armour instead?

Rerolls don’t change the range of possible results (i.e. 1-6) but they do change odds of a particular result.  I’m following the usual format for the percentage odds here so I won’t labour the discussion with what each row means.

1 2 3 4 5 6
= 2.78% 8.33% 13.89% 19.44% 25.00% 30.56%
< 0.00% 2.78% 11.11% 25.00% 44.44% 69.44%
> 97.22% 88.89% 75.00% 55.56% 30.56% 0.00%
<= 2.78% 11.11% 25.00% 44.44% 69.44% 100.00%
>= 100.00% 97.22% 88.89% 75.00% 55.56% 30.56%

Well actually … it is a little different this time so perhaps I should give some explanation to the more interesting bits.

  • The ‘equals’ row gives the odds of a particular result if you were to roll two dice and pick the highest.
  • The ‘less than’ row shows the odds of failing a particular result, e.g. theres a 25% probability of failing a 4+ (rerollable).
  • The ‘greater than or equal to’ row shows the odds of succeeding on a particular result, e.g. the odds of succeeding in a 5+ (rerollable) is 55.56%

“So what?” says you.  In isolation it’s not that insightful, but if we compare it to the odds of a ‘normal’ (i.e. not re-rollable) outcome it gets very interesting.  The odds for a normal roll are covered here, but I’ve graphed the odds of success for normal vs rerollable to really show the difference.

Ok so the odds of getting a 1 or better is 100% in both cases … duh.  But what about that Warlock from the beginning?

The jetbike Warlock has a 33.33% chance of failing that save (with no reroll).  If the Farseer has done his job and put Fortune on that Warlock, then the reroll brings the probability of failure down to 11.11%.  The reroll makes him three times less likely to fail! and what about that terminator armour? it has a 16.67% chance of failing that save – that’s twice as good as the basic Warlock, but one and a half times worse than the Fortuned Warlock!

A fortuned jetbike Warlock has significantly better saving throws than a Terminator with a Stormshield – think about that before you try gunning down a Seer Council.

Take a moment to really look at the chart.  The rerollable 6+ is almost as good as a basic 5+, and is about twice as likely as a normal 6+.  The 5+ reroll is better than a basic 4+, and so on until they converge at 1.  Also notice those trendlines I’ve added, the basic rolls get linearly less likely, but the rerolls stay high and then gradually fall off as you hit 6.

Of course, not all re-rolls are there to mitigate failure – sometimes you can be forced to reroll a success, but that analysis will have to wait for next time…

6 Comments   |  tags: , , , , , , , , | posted in Probability

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