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shotguncoffee
14-10-2012, 00:53
Question 1:
Hacking at someone is a series of interlinked dice rolls:

To hit --> To wound --> (Armour Save) --> (Ward Save)

Now, let's that the model has NO armour save and a 6+ ward save.

Question: Could you move the 6+ save all the way to the front, rolling with -1 to hit instead?
Is this the same probability?


If no, why not?
How are the two probabilities different?


------------

Question 2:

I had asssumed that -1 to hit was just = a flat -1 to hit.

But someone recently took pardon with me, saying that if that -1 to hit goes from hitting on 3+ to 4+, then that's not as big of a deal as if it goes from 5+ to 6+. What he was saying that -1 to hit is not a "flat" nerf but vaxes in power compared to what that roll is.

So though it would be "intuitive" to assume that -1 to hit is just a flat nerf, it is in fact not.

Is this true?

Thanks!

MarcoSkoll
14-10-2012, 01:40
Question: Could you move the 6+ save all the way to the front, rolling with -1 to hit instead?
Is this the same probability?
No. One is multiplicative probability, the other is additive probability.

Let's take a simplified example and say there are two rolls. A 4+ to hit, then a 4+ to wound, with no saves.

Here, we expect one attack in four to wound. Of four, two on average will hit, then one of those two will wound.

If we took that 4+ to wound and applied it as a -3 to hit, then we'd have to roll at least a 7 to hit. Without the rule about "more than 7" (which we are ignoring for the sake of simplicity), this is impossible on a D6. No hits out of four would wound.


So though it would be "intuitive" to assume that -1 to hit is just a flat nerf, it is in fact not. Is this true?
This is the vice versa - a case where probabilities add instead of multiplying them.

If we once again ignore the "7 or more" rule, let's say that a model needs 6+ to hit before a -1 modifier. They now need 7+ and can't hit.
If they need 0+ to hit before that same -1 modifier, they now need 1+ and their chances haven't changed.

In the former case, the modifier is an infinite disadvantage (they've lost all of their chance to hit), and in the other, none at all.

As a financial equivalent, imagine I steal 10,000 (or a value of whatever appropriate currency that represents the equivalent of a small hatchback car) from you. If you own a multinational software company, this a fairly trivial loss. If you're a 9 to 5 guy who works sales for that software company, this is devastating.

~~~~~

Basically, adding probabilities (50% - 50% = 0%) is not the same as multiplying them (50% x 50% = 25%). Which is also why companies like to talk about their "double savings" sales, where they give you 20% off, then another 20% off (or something similar). It's to try and trick you into thinking it's a bigger saving than it is - a lot of people would think of that as 40% off, but it's really only 36% from the original price:

100 product.
Take a fifth off 100 (20) to get 80.
Take a fifth off 80 (16) to get 64.

Total saving: 100-64 = 36.

When you've got a sequence where the probabilities are not mutually exclusive (as successfully hitting and successfully wounding aren't - a hit can fail to wound) and a fraction only applies to the fraction that's made it through the previous stages, probabilities are multiplicative.
When the probabilities are mutually exclusive (such as rolling a 3, and rolling a 4+. You can't do both simultaneously on the same die), that's additive.

de Selby
14-10-2012, 01:52
yeah, what Marco said.

The 6+ ward save is 'flat'; if you get it, it reduces the number of wounds suffered by 16.666 %, no matter how many wounds that is. A 1 pip modifier is not flat: it has a different percentage effect depending on what the original chances of success were.

Etienne de Beaugard
14-10-2012, 01:55
On Question 2, if you look at it from a percent change standpoint, things are different.

% change = (new value - original value)/(original value)

So, a -1 to hit for a 3+ produces

(3/6 - 4/6)/(4/6) = -1/4 or 25% decrease.

but the same -1 to hit for a 4+ produces
(2/6 - 3/6)/(3/6) = -1/3 or ~33% decrease

drat! Ninja'd.

shotguncoffee
14-10-2012, 04:13
Thanks for all the help and replies, guys!

Ok. Lets say we introduce the rule that:

A natural 1 always fails
A natural 6 always succeeds


Is 6+ ward save then the save as -1 to hit? - which is better/worse?
In another example, say you have hit and wound. (No ward save or armour save.) - is -1 S and -1 to hit the same then?

Scaryscarymushroom
14-10-2012, 04:38
Is 6+ ward save then the save as -1 to hit? - which is better/worse?
No, they aren't the same in this instance either.

Let's say you have a 2+ to hit, and for the sake of convenience, you wound automatically. That's 83% chance to hit. And then there is a 6+ ward save, meaning there's a 17% chance to save (or an 83% chance the of a casualty).

Let's say you roll 120 dice at a 2+. 100 hits. You wound automatically. Then you scoop up those 100 hits and roll 6+ saves. 83 casualties.
Now let's say you have 120 dice and you roll a 3+ wounding automatically and no ward saves. You get 80 casualties.

You have better chances of causing a casualty with the 2+/6+ than you do with the 3+/nil.


In another example, say you have hit and wound. (No ward save or armour save.) - is -1 S and -1 to hit the same then?

It depends. Take a 4+ to hit 4+ to wound, for example.

1/2 of the shots will hit, and 1/2 of those will wound. So you have a 25% chance to inflict a casualty.
Change it to a 5+ & 4+, and you get 1/3 will hit and 1/2 of those will wound. 1/3*1/2=1/6.
Change it to a 4+ & 5+, and you get 1/2 will hit and 1/3 of those will wound. 1/2*1/3=1/6. No difference.

But if you have a 3+ to hit and a 5+ to wound, there will be a big difference.
2/3 will hit, and 1/3 of those will wound. 2/3*1/3=2/9. Which is roughly 22%.
Change it to a 4+, 5+, and you get 1/2*1/3=1/6. Roughly 17%. BUT
Change it to a 3+, 6+, and you get 2/3*1/6=1/9. Roughly 11%. Not the same.

So you could swap the to hit/to wound rolls and the statistics work out the same, but you can't scale one up by 1 and the other down by 1 unless that means that they would swap (e.g. 3+, 4+ becomes a 4+, 3+. That's the same. Likewise, a 2+, 6+ is the same as a 6+, 2+. But a 3+, 4+ does not equal the same thing as a 2+, 5+.)

---

I hate to say it, but the easiest way is to just roll the dice the game tells you to roll. :/
If you want to speed things up, you can roll several dice at once and check them all. Like, one blue (hit), one green (wound), one red (save), and one white (ward). If the white one comes up a 6, it doesn't matter what the other rolls are. Likewise, if the blue one misses or the green one fails to wound, it doesn't matter what the other ones say. But this only works if you're rolling a single hit-wound-save-ward chain. If you need to roll 2 or more dice to hit, the only way you'll get an accurate read is by doing it the old fashioned way. Or using an app. :shifty:

Verm1s
14-10-2012, 11:21
If you have a mathhammer question then yes, you do need help.

Hendarion
14-10-2012, 13:03
100 rolls 4+ == 100 * 1/2
100 rolls 5+ == 100 * 1/3
100 rolls 4+ with 6+ ward == 100 * 1/2 * 5/6 == 100 * 5/12 =/= 100 * 1/3

MarcoSkoll
14-10-2012, 13:28
Ok. Lets say we introduce the rule that: A natural 1 always fails, A natural 6 always succeeds.
Is 6+ ward save then the save as -1 to hit? - which is better/worse?
To reuse my original example of 4+ to hit and 4+ to wound:

As before, we expect one attack in four to wound. Of four, two on average will hit, then one of those two will wound.

If we took that 4+ to wound and applied it as a -3 to hit, then we'd be hitting on 6s only, with autowounds. Four attacks, of which one in six hits - and four divided by six is clearly less than one.

Under most circumstances, when reducing probabilities, additive probabilities have a greater effect than multiplicative ones. (See the "double savings" example I gave, the difference between 40% savings and 36% savings.)

So, -1 to hit is usually going to be better than a 6+ ward. If it was normally 5+ to hit, -1 would halve the chance to hit to just 6, halving the likely number of wounds - whereas a 6+ ward would only reduce it by a sixth.

However, in the cases of "1 or less" or "6 or more" to hit, then that chance wouldn't be affected by a -1 (they'd still need a 2+ or a 6), so the 6+ ward would be more advantageous here. And of course, when mentioning template weapons for which a -1 to hit would be moot.

I'd say -1 was more use normally... but there are some circumstances in which it would have no effect. Whereas a 6+ ward is always of SOME use.


In another example, say you have hit and wound. (No ward save or armour save.) - is -1 S and -1 to hit the same then?
See Scaryscarymushroom's post.

shotguncoffee
15-10-2012, 02:30
thanks everyone, you've been much help indeed!

Sgt John Keel
15-10-2012, 02:54
yeah, what Marco said.

The 6+ ward save is 'flat'; if you get it, it reduces the number of wounds suffered by 16.666 %, no matter how many wounds that is. A 1 pip modifier is not flat: it has a different percentage effect depending on what the original chances of success were.

I find it curious that no one mentioned the concept of percentage points by name.

Hendarion
15-10-2012, 08:38
That is most likely because the term isn't understood by that many people, especially when they are already asking for others to do the math for them.

Sgt John Keel
15-10-2012, 13:35
That is most likely because the term isn't understood by that many people, especially when they are already asking for others to do the math for them.

Yes, obviously. I did not mean or intend the name to be the sole explanation, merely that naming (a concept pertinent to) what they all have explained so nicely might help the OP re-thread (and recognise) the subject in the future if so desired.

Hendarion
15-10-2012, 14:07
Ah, indeed. :)

MarcoSkoll
15-10-2012, 14:14
I did think about mentioning percentage points, but made the same reasoning as Hendarion. For someone rusty on statistics and fractions, it is quite a lot to suddenly be talking about two different types of percentage - hence why I talked about it in terms of adding or multiplying probabilities.

Little Wolf
23-10-2012, 13:11
And then there's the fact that for non-native speakers like me, mathematical english is pretty hard. My english is fine, my maths is fine, but my mathematical terminology sucks...

Let alone the fact that 3,403 in my 'native maths' is 1/1000 of 3,403 in 'english maths'. (We pretty much do the , and the . the other way round.) So some signs are different too.