Ruin


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Ruin is a warlock talent, considered to be the cornerstone of the Destruction talent tree, that increases the critical hit damage bonus of your Destruction spells by up to 100%. A normal spell critical is +50% damage taking into account all other damage modifiers. Ruin doubles the critical bonus when maxed, so you get a +100% damage bonus on critical hits.

Long term damage
This is a formula that calculates how much of a boost (over a large number of spell casts) in damage you can expect to get from this talent when using spells that can crit. The formula uses the "three outcomes" model, as well as a concept from probability theory called expected value. This calculation does not take into account resists and partial resists.

Mathematical derivation
The "three outcomes" model states that in WoW, when one casts a spell there are three possible results: a hit, a critical hit (crit), or a miss. If we know the probability of each outcome, we can use probability theory to calculate the expected value of damage. Then, we can calculate the percentage difference in damage we can expect to see from various crit rate/hit rate combinations.

&#91;(chance to crit) &times; (crit damage)&#93; + &#91;(chance to hit and not crit) &times; (average damage)&#93; + &#91;(chance to miss) &times; 0&#93;

Variables

 * C = Chance to get a crit.
 * H = Chance to hit.
 * A = Average spell damage (including all damage modifiers).
 * 1.5&times;A = Normal crit damage.
 * 2&times;A = Crit damage due to Ruin.
 * (1-H) = Chance you will miss (and do 0 damage).
 * (H-C) = Chance to hit and not crit.

Algebra
Without Ruin, the expected amount of damage one does with spells is:

C&times;1.5&times;A + (H-C)&times;A + (1-H)&times;0

With Ruin, one expects:

C&times;2&times;A + (H-C)&times;A + (1-H)&times;0</tt>

The percentage difference would be:

( (C&times;2&times;A+(H-C)&times;A) - (C&times;1.5&times;A + (H-C)&times;A) ) / (C&times;1.5&times;A + (H-C)&times;A)</tt>

This expression simplifies to:

C/(C+2H)</tt>, where 0 ≤ C ≤ 1</tt>; 0 ≤ H ≤ 1</tt>; and (H-C) ≥ 0</tt>

Results
Essentially, this result states that the increase in long term damage you get from this talent depends on two variables. The first variable is C, your crit rate. The other variable is H, your chance to hit.

The role that your chance to hit plays may seem counter-intuitive (i.e. lower hit rates leads to higher bonus). This is a consequence of the "three outcomes" model. Crits and hits share the same "probability space", and therefore hit rate puts a cap on crit rate. It must be thought of in this way: if you are missing more often, then the critical hit to "normal" hit ratio is higher.

Let's say one has a 5% chance to crit.


 * Scenario 1: If you have a 1% chance to miss, then you have an 99% chance to hit. Therefore, 5/99 is your chance to critically hit, when you hit.


 * Scenario 2: If you had a 95% chance to miss, then you have a 5% to hit you'll have a 5/5 chance to crit if you don't miss. Keep in mind that you are missing 95% of the time.  In this special case, Ruin will give you a nice bonus compared to a character that doesn't have it.  Chances are that this character will be dead before he can witness his "mad Ruin crits".


 * Scenario 3: If you have a 99% chance to miss, then you have a 1% chance to hit. You have a 5% chance to crit, but only a 1% chance to hit.  It is unclear how the game would react to this situation.  It is likely that misses take priority over crits when the percentages overlap.  From a practical point of view, it would not matter because you are probably not going to win the encounter.  The best course of action would be to use a Swiftness Potion and run away!

It would be foolish to pass up +hit gear and use the excuse that it will ruin Ruin. One has to realize that if you are missing more often, you are doing less damage. Overall, Ruin is a nice 1 point talent that gives a nice DPS increase to spells that can crit. A 21.333% crit rate, and a 96% hit chance will match the 10% bonus effect that 5 points in Shadow Mastery has on shadow spells.

Examples
Against enemies that are the same level as you, you have a 96% base chance to hit.


 * 5% crit rate, 96% chance to hit
 * 0.05 / (0.05 + 2*(0.96)) = 0.025641...</tt>, or about +2.6% expected damage


 * 10% crit rate, 96% chance to hit
 * 0.10 / (0.10 + 2*(0.96)) = 0.04950...</tt>, or about +5% expected damage


 * 21.3333% crit rate, 96% chance to hit
 * 0.213333 / (0.213333 + 2*(0.96)) = 0.1</tt>, +10% damage (about the same as 5 points in Shadow Mastery).


 * 20% crit rate, 80% chance to hit
 * 0.20 / (0.20 + 2*(0.80)) = 0.11...</tt> or 11% boost in damage, but DPS is lower because you miss more often.


 * 5% crit rate, 5% chance to hit
 * 0.05 / (0.05 + 2*(0.05)) = 0.33...</tt> or 33% bonus. Counter-intuitive, but remember that you're missing 95% of the time.  You are also critting every time you hit because crits and hits overlap.  You will likely never experience this in the game.