Theorycrafting and Simple Math

If you follow me on twitter or spend any time in #Ravenholdt you’ve probably heard one of my “theorycrafting doesn’t require complex math” rants.  Recently someone challenged me in IRC to prove that you could do useful stuff with basic math so in this post I will show that you can answer useful theorycrafting questions with just simple math.

 

First we need to define simple math.  Occasionally when I make this argument people accuse me of underestimating complexity because I like math. To answer this objection I will clearly define what I mean by simple math.  First simple math includes basic arithmetic, addition, subtraction, multiplication and division.  Second simple math includes basic probability concepts.  To make sure I’m not cheating here and smuggling in complex math under the guise of “basic probability” I’m going to start with four probability questions.  Anyone who can answer all four of these questions possesses what I am calling basic probability knowledge for the purposes of this post.

{slider Question 1) If you roll a six sided die once, what is the probability that you will roll a 6?|closed|noscroll}1/6 = 0.166 = 16.6%{/sliders}
{slider Question 2) If you roll a six sided die once, what is the probability you do not roll a 6?|closed|noscroll}1-(1/6) = 5/6 = 0.833 = 83.3%{/sliders} 
{slider Question 3) If you roll a six sided die twice what is the chance you get two 6s?|closed|noscroll}(1/6)*(1/6)=(1/36)=0.027= 2.7%

Concept: The probability of two events happening is the product of the probabilities of each event (Technically this only applies to independent probabilities only but in WoW most probabilities are independent).{/sliders}
{slider Question 4) In a lottery you roll a six sided die once, if you roll a 6 you get $10, if you roll a 5 you get $5, and if you roll a anything else you get nothing.  If you enter this lottery 100 times how much do you expect to win?|closed|noscroll}Answer: First work out the probabilities of each event.  From question 1 we know that the probability of rolling a 6 is 1/6 and the probability of rolling a 5 is 1/6.  From question 2 we know probability of rolling anything else is 1-(1/6)-(1/6)=4/6=0.666=66.6%

We’re entering 100 times so we expect to roll 16.6 “6s”, 16.6 “5s” and 66.6 other.  Now multiply each by the associated winning, 16.6*$10=$166, 16.6*$5=$83.  So the total expected winnings from 100 plays is $166+$83=$249 and if we divide that by 100 we get $2.49 dollars per roll (Some people may note that $2.50 is actually the correct result which we didn’t get due to rounding but that isn’t really important to the concept on display here).

Concept: The average value of an event with multiple possible outcomes is the sum of the value of each outcome times the probability of that outcome occurring.{/sliders}

Armed with basic arithmetic and these four probability concepts we’re going to compute a useful an actionable theorycrafting result, the crossover point where Anticipation becomes better than Marked for Death (MfD) in terms of combo points saved/generated per minute.

To begin a few assumptions:
1) 100% Revealing Strike (RvS) uptime but we will not consider actually casting RvS.
2) Per traditional rotations, only use finishers at 5 combo points, this also allows us to assume the player always starts with 1 combo point from Ruthlessness.
3) This does not consider using anticipation to shift finishers to higher levels of Bandit’s Guile.
4) This analysis considers patchwork dps with no opportunities for MfD cooldown reset.
5) No set bonuses, modifying this analysis for set bonuses is left as an exercise for the reader.

To find the crossover point we need to determine how many combo points are wasted when using MfD instead of anticipation.  There are a number of ways to handle this but for simplicity we’re just going to write out all the paths from 1 combo point (see assumption 2) to 5 combo points and note which paths do and do not have waste.

a) 1->2->3->4->5
b) 1->2->3->4->6 (Waste)
c) 1->2->3->5
d) 1->2->4->5
e) 1->2->4->6 (Waste)
f) 1->3->4->5
g) 1->3->4->6 (Waste)
h) 1->3->5

Now we need to work out a probability of each sequences. From the Revealing Strike tooltip we know that 25% of the time (0.25) we generate 2 combo points and from question 2 we know that we have a 1-0.25 =0.75 or 75% chance of generating one combo point.  This makes working out the probability of each sequence above is an application of question 3.

a) 0.75*0.75*0.75*0.75= 0.31640625
b) 0.75*0.75*0.75*0.25= 0.10546875
c) 0.75*0.75*0.25= 0.140625
d) 0.75*0.25*0.75= 0.140625
e) 0.75*0.25*0.25= 0.046875
f) 0.25*0.75*0.75= 0.140625
g) 0.25*0.75*0.25= 0.046875
h) 0.25*0.25= 0.0625

If we sum the probabilities of the three paths that lead to waste we get a probability of wasting a combo point due to a badly timed proc of 0.10546875+0.046875+0.046875= 0.19921875 or 19.92%.

This is a relatively interesting result on its own but it isn’t an actionable result yet.  To create a result that informs our play we need to go one step further.  We know MfD generates 4 combo points per minute and we just derived that anticipation saves 0.1744 combo points per finisher.  So how many finishers does a player need to use before anticipation saves more combo points than MfD generates? 

4 combo points per minute/0.19921875 combo points per finisher= 20.08 finishers per minute.  We could this analysis further and look whether in your gear you can reach that crossover point but in this case it is probably safe to assume you’ll never be using more a finisher every 3 seconds and leave it there. This isn’t a new result, this is something that all of our theorycrafting tools tell us, MfD is better than anticipation.  However this analysis doesn’t require us to break out those big complex tools to answer the question. A relatively simple analysis requiring only “simple math” can provide the same answer.

So what am I trying to say here? I’m not trying to tell you that theorycrafting is easy or that theorycrafting is for everyone.  Figuring out how to model mechanics is often quite challenging. What I hope people get out of this post is, don’t be intimidated by the math. Too many times I read posts on the forum or comments in IRC saying, “I’d love to help theorycraft but I’m bad at math” and talk themselves out of contributing.  What this post shows, I hope, is that you don’t need to be “good at math” to contribute in a useful, meaningful way to WoW theorycraft.