Dharok's: A Theory, A Formula, A Dream.

[Boa1891]

Now, I often hear people say that Dharok's is less effective than the whip-speed weapons unless it can hit almost double consistently. Is that really the case?
Quick numbers: The whip takes 4 server cycles (2.4 seconds) per attack, and the greataxe takes 7 (4.2 seconds). Taking only this into account, the Greataxe's average hit "must" be 7/4 of the whip/saradomin sword's average hit in order for it to be equally effective. Taking into account the fact that the Greataxe is more accurate than either of them, there is a little leeway produced allowing the greataxe to hit a smidgen lower and still be equal.
While Dharok's meets this requirement very well at it's maximum efficiency, why is it that field testing shows it's better per hour even in cases where the max hit is ~80, when the whip/saradomin sword can cleanly max 44-45 in the same circumstances?

Let's take waterfiends as an example, with 128 HP and near-immunity from everything but Crush. In this example, let's say the sara sword's max is 44 and the dharok's axe is 7/4 of 44, which is 77. They should overall be equally efficient.
First, how long it will take to kill them if you max with each hit:
(NOTE: The lag-time after the kill is not counted, and the first hit takes no time at all)
3 hits with the sara sword, 4.8 seconds
2 hits with dharok's axe, 4.2 seconds

If you hit what weapons "average", their mid-hit:
Sara sword: 22-22-22-22-22-18 (6 hits, 12 seconds)
Dharok's axe: 38-39-38-12 (4 hits, 12.6 seconds)


So what's the breaking factor making these two sets of data average equal in time?
The Knock-out hit.
In order for this particular set-up to be even, INCLUDING post-kill lagtime, the Dharok's greataxe must be able to two-hit kill waterfiends as often as the sara sword 3 hit OR 4 hits them. I think I've come up with a very clear way of figuring out the chance of a certain x-hit KO's likelihood, excluding misses (that does not mean excluding 0s).
With dharok's axe in this scenario, if you launch two hits at a waterfiend you can do 0-154 damage. Since they have 128 HP, and every possible hit has the same likelihood, there is a 26/155* chance to 2-hit kill, that's 16.8%.
Now, the Sara Sword. If you Launch 3 hits you can do up to 132 damage; with four hits, up to 171**. You average these two likelihoods together:
4/133 and 44/172 average together to equal 14.3%.
This means that EXCLUDING the additional accuracy Dharok's axe has, if Dharok's max hit is 7/4 of another option's max it's overall more efficient.

*This is out of 155 rather than 154 because 0 is a possible outcome, along with 154 numbers.
**You will only hit a fourth time if you deal 127 or less HP in three hits, meaning the max for four hits is 171 rather than 176.

I tried to do these calculations without going too in-depth on how the KO-hit affects them, but in order to understand why this math works you have to know a bit about how a hit is calculated.
Whenever you're attacking something, it first checks goes through a "hit or miss" check, that gauges your attack style, level, and bonus against that of your foe's defence using a formula and a random number generator to determine whether you hit or miss. If you miss, hit a 0. If you hit, move on to the next check.
It next generates a random number from 0 to your max hit, which is floored. This is your hit. (As such, it is entirely possible to "hit" a 0; not all 0s are misses.)
This hit is applied to the enemy no matter what their HP is, however it is limited by their HP. For example, if a monster has 20 HP and your max is 20, there is a 1/21 chance you'll hit 20 to KO them (assuming the only 0 possible is a hit 0, not a miss 0). However, if your max hit is 30, there is a 11/31 chance you'll hit a 20 and KO it.
Thus, kills per hour for the larger hits with slower speed are slightly better than the smaller hits with higher speed, as long as Damage per Second is similar. Since higher hit weapons generally have a greater accuracy, this is safe to say.

If anyone thinks they see issues in my math, please bring them up. I love math battles, and I always learn from my mistakes if someone shows me up.

Fun Fact: Your max hit is the result of a formula which does, in fact, output a decimal. This decimal IS in fact multiplied by the random number generator without being floored first, which means that your absolute max hit will always have a lower probability of being produced than all the other hits. The difference in probability is equal to x where x is (max hit floored+1)-(max hit formula output).
For example, (42+1)-(42.2), meaning it is 80% less likely to appear than any other number.
Note that if you wish to collect data on the probability of each hit to check this, you must omit any hit that kills the foe.

[Blizaga57]

3 hits with the sara sword, 4.8 seconds
2 hits with dharok's axe, 4.2 seconds

I'll keep looking, but for now the only glaring flaw is that you assume either you're killing only one (unrealistic) or that you can attack the next monster instantly after the first is dead.

Lets kill 10 waterfiends, maxing every time, missing no attacks. Also assume the first hit has no delay time. t = 0.6 seconds, one server cycle.

Dharoks: 20 attacks = (20-1)*7t = 133t
Sara Sword: 30 attacks = (30-1)*4t = 116t

In this case, the sara sword easily beats dharok's.

[Doomedrusher]

Dharoks is a placebo; "OMG BIG HITS". It's basically like the AGS; people prefer to use the AGS over the BGS, even though the BGS gives them a higher expected return per special bar.

This is going in the index anyways.

[Boa1891]

3 hits with the sara sword, 4.8 seconds
2 hits with dharok's axe, 4.2 seconds

I'll keep looking, but for now the only glaring flaw is that you assume either you're killing only one (unrealistic) or that you can attack the next monster instantly after the first is dead.

Lets kill 10 waterfiends, maxing every time, missing no attacks. Also assume the first hit has no delay time. t = 0.6 seconds, one server cycle.

Dharoks: 20 attacks = (20-1)*7t = 133t
Sara Sword: 30 attacks = (30-1)*4t = 116t

In this case, the sara sword easily beats dharok's.

(NOTE: The lag-time after the kill is not counted, and the first hit takes no time at all)
3 hits with the sara sword, 4.8 seconds
2 hits with dharok's axe, 4.2 seconds
.........
In order for this particular set-up to be even, INCLUDING post-kill lagtime, the Dharok's greataxe must be able to two-hit kill waterfiends as often as the sara sword 3 hit OR 4 hits them.

The first numbers stated that I intentionally left that out, but in my later calculations I DID factor them. The later calculations were the important part, as shown by your figures which very clearly ignored the points presented and thus gave an incorrect calculated return.

[Blizaga57]

My bad. After I read it the first time, I went through again, just looking at the numbers to see what I could find.

I'd imagine that the ko hit has a diminishing effect as the average hit is lowered from the max to 50% of the max to a realistic number (40-45%).

My mind keeps thinking about the missing penalty, and how it's much longer for dharok's than ss, but I don't think there's anything there.

Very nice work on the creation of an "x hit" ko system.

[Craven Range]

Can you please explain this to me?

This hit is applied to the enemy no matter what their HP is, however it is limited by their HP. For example, if a monster has 20 HP and your max is 20, there is a 1/21 chance you'll hit 20 to KO them (assuming the only 0 possible is a hit 0, not a miss 0). However, if your max hit is 30, there is a 11/31 chance you'll hit a 20 and KO it.

So.. In lamens terms

Monster A: 20hp

Player A: Max hit of 20

if a monster has 20 HP and your max is 20, there is a 1/21 chance you'll hit 20 to KO them

Therefore.. a 4.7% Chance to K.o

Player B: Max hit of 30

However, if your max hit is 30, there is a 11/31 chance you'll hit a 20 and KO it.

Therefore.. a 35% Chance to K.o

Is this correct? Im 113 Combat with 78 Strength.. It looks like im cutting myself Very short in terms of training my slayer... By this strength is very important! The task that comes to mind here (I do sumona) is cave crawlers.. they have 22 hits.. I think i better train my strength.. By the looks of things im training alot slower than anyone else my combat level.
Thankyou boa

[wkw427]

Dharok is better then something faster on something with a huge amount of hp. For example, if I hit 70 on a spirit mage, I have to hit a 5 again. Not cool

[Boa1891]

I hate to say it to fans of attack or defence, but for terms of melee (and slayer) training speed strength is THE most important stat- Especially on monsters with lowish defence. Trust me, I have all three 99 I sometimes regret getting 99 attack before strength, because strength would have sped up my attack training considerably.
Yes, you interpreted that correctly.
For example: Go to lumbridge and kill the level 1 spiders. You'll almost always kill them in one hit, yeah? That's because the random number generator will almost always give you a hit higher than their HP (2?). When you eventually hit a 0 or 1 on one, it makes you feel REALLY weak.

Now imagine it in the opposite sense. If your max is 20 and you're killing something with 20HP, only one out of every 21 kills (remember, this is assuming you hit and not miss) will be a one hit kill. Once you ruin the chance for a one hit kill, the next hit couldn't be a one hit kill, meaning that for every 1 hit kill there are 20 2+ hit kills (on average), which is 40 hits, half of which have had their EXP return diminished by having their max hit cut short.

Imagine if you have a max hit of 45- There is a more than 50% chance of 1-hit killing the monster with 20HP!

[Doomedrusher]

And the XP return is STILL diminished because you hit a 20 when you could be hitting a 45.

[nescap]

I hate to say it to fans of attack or defence, but for terms of melee (and slayer) training speed strength is THE most important stat- Especially on monsters with lowish defence. Trust me, I have all three 99 I sometimes regret getting 99 attack before strength, because strength would have sped up my attack training considerably.
Yes, you interpreted that correctly.

So if you had two players:
Player A: Att-70, Str-99, Def-70
Player B: Att-99, Str-70, Def-70

Player A would train faster because each time they hit, they would have larger potential damage?

But I wonder how much the difference in Attack would affect the hit/miss calculator? You don't think Player B would train faster because they would have less misses (even though their max hit was less)?

[wkw427]

I say that when two players atk and str are =.. Wait

P1 = 70 atk 99 str
P2 = 99 atk 70 str

P1 atk = P2 str
P1 str = P2 atk

P1 hits less often, but higher. Say he misses 50% of the time but always hits 30

P2 hits more often, but less. Say he hits 100% of time but always hits 15

Training speed = same

[Qeltar]

Well, too tired to read through all the math. All I know is that regardless of the theory, if you actually measure kill rates, Dharoks is the fastest -- assuming you use Piety, which you should anyway.

[FinsToTheLeft]

I hate to say it to fans of attack or defence, but for terms of melee (and slayer) training speed strength is THE most important stat- Especially on monsters with lowish defence. Trust me, I have all three 99 I sometimes regret getting 99 attack before strength, because strength would have sped up my attack training considerably.
Yes, you interpreted that correctly.
For example: Go to lumbridge and kill the level 1 spiders. You'll almost always kill them in one hit, yeah? That's because the random number generator will almost always give you a hit higher than their HP (2?). When you eventually hit a 0 or 1 on one, it makes you feel REALLY weak.

Now imagine it in the opposite sense. If your max is 20 and you're killing something with 20HP, only one out of every 21 kills (remember, this is assuming you hit and not miss) will be a one hit kill. Once you ruin the chance for a one hit kill, the next hit couldn't be a one hit kill, meaning that for every 1 hit kill there are 20 2+ hit kills (on average), which is 40 hits, half of which have had their EXP return diminished by having their max hit cut short.

Imagine if you have a max hit of 45- There is a more than 50% chance of 1-hit killing the monster with 20HP!

Strength probably wouldn't make a large difference on higher health monsters, like the ones assigned my Duradel. Besides, not using controlled on the whip means that one must use a Saradomin sword to train strength.

[Lord Klotski]

Boa1891 wrote: With dharok's axe in this scenario, if you launch two hits at a waterfiend you can do 0-154 damage. Since they have 128 HP, and every possible hit has the same likelihood, there is a 26/155* chance to 2-hit kill, that's 16.8%.

Ahh...I get a chance to correct your math. I won't pass this up.

I will argue that the damage from 0-154 will not be uniformly distributed: that is, I argue that every possible hit from 0 - 154 does not have the same likelihood.

Let's consider a very simple situation, that will explain this problem clearly. Consider a player with a max hit of 6. He attacks twice. Assume that he hits red splats (does not miss) both times, does not hit 0's , whatever you want to call it. We can model this situation with a pair of normal dice.

As you likely know, getting a 2 has a probability of 1/36, getting a 7 has a probability of 6/36, etc. The end result of adding the two dice is biased towards the center. Even though we start out with a discrete uniform distribution, adding the two dice together looks a lot more like the normal distribution. This is due to the central limit theorem, for the record.

So here, we have two dice that "roll" from 0 - 77, assuming that there are no "misses", per se. If you want, you can construct a table by hand. That is extremely painful. We can do better, methinks.

There are 78^2 = 6084 outcomes. We are interested in the ones that have a combined sum of greater than 127, greater than or equal to 128, whatever floats your boat.

I'll call you back to the "standard dice example" for the easiest way to solve this. If you want to know the outcomes on two dice , you make a square chart that looks like THIS: http://education.yahoo.com/homework_hel ... 7-ex-1.gif. On the top and side are the two dice in question, in the middle 36 squares are the numbers that you get from combining the two dice. Note, for instance, that the numbers greater that or equal to 10, say, are in a triangle shape. This is important.

Keeping the idea of "the triangle" in mind, it's actually trivial to calculate the total outcomes greater than 128. 128-77 = 51. 77-51 + 1 is the "base of the triangle" - width of 27. Now this is a "discrete triangle" , so the area is 27+26+25+24....+1. This is easily calculated also: 27*28/2 (a formula from discrete mathematics) - gives 378.

I propose that the odds of dharoks double hitting a waterfiend is 378/6084 (6.2%).

Since I don't do things by halves, I simulated 3549 kills. 201 were double hits. That's 7.9%. Close enough, I guess. Not exactly 16% though.

[J Gard]

I can't wait to reread this after I have eaten, gotten some sleep, etc...

I will then probably be able to almost completely understand it then
instead of the about 50% I can now

Curse this thing that they call school

[FinsToTheLeft]

Boa1891 wrote: With dharok's axe in this scenario, if you launch two hits at a waterfiend you can do 0-154 damage. Since they have 128 HP, and every possible hit has the same likelihood, there is a 26/155* chance to 2-hit kill, that's 16.8%.

Ahh...I get a chance to correct your math. I won't pass this up.

I will argue that the damage from 0-154 will not be uniformly distributed: that is, I argue that every possible hit from 0 - 154 does not have the same likelihood.

Let's consider a very simple situation, that will explain this problem clearly. Consider a player with a max hit of 6. He attacks twice. Assume that he hits red splats (does not miss) both times, does not hit 0's , whatever you want to call it. We can model this situation with a pair of normal dice.

As you likely know, getting a 2 has a probability of 1/36, getting a 7 has a probability of 6/36, etc. The end result of adding the two dice is biased towards the center. Even though we start out with a discrete uniform distribution, adding the two dice together looks a lot more like the normal distribution. This is due to the central limit theorem, for the record.

So here, we have two dice that "roll" from 0 - 77, assuming that there are no "misses", per se. If you want, you can construct a table by hand. That is extremely painful. We can do better, methinks.

There are 78^2 = 6084 outcomes. We are interested in the ones that have a combined sum of greater than 127, greater than or equal to 128, whatever floats your boat.

I'll call you back to the "standard dice example" for the easiest way to solve this. If you want to know the outcomes on two dice , you make a square chart that looks like THIS: http://education.yahoo.com/homework_hel ... 7-ex-1.gif. On the top and side are the two dice in question, in the middle 36 squares are the numbers that you get from combining the two dice. Note, for instance, that the numbers greater that or equal to 10, say, are in a triangle shape. This is important.

Keeping the idea of "the triangle" in mind, it's actually trivial to calculate the total outcomes greater than 128. 128-77 = 51. 77-51 + 1 is the "base of the triangle" - width of 27. Now this is a "discrete triangle" , so the area is 27+26+25+24....+1. This is easily calculated also: 27*28/2 (a formula from discrete mathematics) - gives 378.

I propose that the odds of dharoks double hitting a waterfiend is 378/6084 (6.2%).

Since I don't do things by halves, I simulated 3549 kills. 201 were double hits. That's 7.9%. Close enough, I guess. Not exactly 16% though.

Does experience in the field confirm your theory?

[Boa1891]

Strength probably wouldn't make a large difference on higher health monsters, like the ones assigned my Duradel. Besides, not using controlled on the whip means that one must use a Saradomin sword to train strength.

I was talking in terms of exp per hour; since you rarely if ever miss on a majority of slayer monsters, a higher hit is more helpful than added accuracy that essentially goes into the abyss.

Boa1891 wrote: With dharok's axe in this scenario, if you launch two hits at a waterfiend you can do 0-154 damage. Since they have 128 HP, and every possible hit has the same likelihood, there is a 26/155* chance to 2-hit kill, that's 16.8%.

Ahh...I get a chance to correct your math. I won't pass this up.

I will argue that the damage from 0-154 will not be uniformly distributed: that is, I argue that every possible hit from 0 - 154 does not have the same likelihood.

Let's consider a very simple situation, that will explain this problem clearly. Consider a player with a max hit of 6. He attacks twice. Assume that he hits red splats (does not miss) both times, does not hit 0's , whatever you want to call it. We can model this situation with a pair of normal dice.

As you likely know, getting a 2 has a probability of 1/36, getting a 7 has a probability of 6/36, etc. The end result of adding the two dice is biased towards the center. Even though we start out with a discrete uniform distribution, adding the two dice together looks a lot more like the normal distribution. This is due to the central limit theorem, for the record.

So here, we have two dice that "roll" from 0 - 77, assuming that there are no "misses", per se. If you want, you can construct a table by hand. That is extremely painful. We can do better, methinks.

There are 78^2 = 6084 outcomes. We are interested in the ones that have a combined sum of greater than 127, greater than or equal to 128, whatever floats your boat.

I'll call you back to the "standard dice example" for the easiest way to solve this. If you want to know the outcomes on two dice , you make a square chart that looks like THIS: http://education.yahoo.com/homework_hel ... 7-ex-1.gif. On the top and side are the two dice in question, in the middle 36 squares are the numbers that you get from combining the two dice. Note, for instance, that the numbers greater that or equal to 10, say, are in a triangle shape. This is important.

Keeping the idea of "the triangle" in mind, it's actually trivial to calculate the total outcomes greater than 128. 128-77 = 51. 77-51 + 1 is the "base of the triangle" - width of 27. Now this is a "discrete triangle" , so the area is 27+26+25+24....+1. This is easily calculated also: 27*28/2 (a formula from discrete mathematics) - gives 378.

I propose that the odds of dharoks double hitting a waterfiend is 378/6084 (6.2%).

Since I don't do things by halves, I simulated 3549 kills. 201 were double hits. That's 7.9%. Close enough, I guess. Not exactly 16% though.

You caught me here, to my surprise. While I had questioned in my mind the accuracy of that particular piece of math, I couldn't put my finger on what was wrong. While every hit has an equal probability of occurring on one hit, I forgot that there were two hits here.

Could you possibly calculate (or even simulate, if it's easier; "close enough" eh?) the likelihood, then, of a 3-4 hit kill using this hypothetical sara sword? I'm sure I could do it myself now that I'm reminded of these bell curve tendencies, but I'm honestly all mathed out for the day. I'm extremely sure that some how, some way, I can mathematically prove that Dharok's > Whipspeed weapons.

Side note: How far along in your education are you? I'm going into my junior year of HS.

Well, too tired to read through all the math. All I know is that regardless of the theory, if you actually measure kill rates, Dharoks is the fastest -- assuming you use Piety, which you should anyway.

I agree for sure.

As a 'more on-topic' side note: I particularly recall one instance where I 2-hit four waterfiends in a row. I was elated. That was the second trip I was using dharok's, and I started right after I read Qeltar's guide. I love it to death, if only I had more money for prayer pots.

[Lord Klotski]

Kk, I'll get a simulation happening for you right now. I'll post results shortly.

As far as education goes, I'm 18 and finished two years of university.

I owed this correction to you, I must say - you've eaten me alive on most everything that I've written lately, so I just had to return the favour.

FWIW, I can even simulate average kill time with the DH and the SS. If you own a TI graphing calculator, I could even explain to you how to write a simple program to do it youself, if you want that instead.

[Boa1891]

Kk, I'll get a simulation happening for you right now. I'll post results shortly.

As far as education goes, I'm 18 and finished two years of university.

I owed this correction to you, I must say - you've eaten me alive on most everything that I've written lately, so I just had to return the favour.

FWIW, I can even simulate average kill time with the DH and the SS. If you own a TI graphing calculator, I could even explain to you how to write a simple program to do it youself, if you want that instead.

In order-
Thanks.

Good to know ^_^

I haven't really corrected you, just added more than you did. You probably left the stuff I did out for brevity or because you couldn't be bothered, with the exception of the encyclopedia.

I don't have a graphing calculator, actually. I'm sure I could write something on my computer to do it, if I tried. If you could calculate average kill time, that's great, but if it comes out with the sara sword on top there must be something I'm not thinking of. Perhaps I'm just using too low a hit from the Dharok's, which would mean my entire point is moot

[Lord Klotski]

Well:

SS testing (simulating)

5000 kills

275 4-hits

1 3-hit

This should come as no particular surprise - think of the odds: You roll three dice, each of which has 45 sides. To get a 3-hit each dice must be averaging 43+. Unlikely.

Oh - get a graphing calculator. Sell that crappy webcam if you have to, it's a worthwhile trade. ROFL.

If you give me numbers for assumed accuracy, I will consider simulating the whole nine yards. Personally, I'll give the SS 70% accuracy with piety and my gear, because that was what I tested when I tested it . Never really had the guts to test the accuracy of DH - I set my calculator up so that I press "1" and it's a hit, "2" and it's a miss..... but I need all my concentration at 13 HP (I ignore Qeltar's ridiculous recommendation to keep health at a sane level )