Darkmoon Card: Wrath

This is one of the four available Darkmoon Cards trinkets added with the Burning Crusade expansion.

Obtaining the Trinket
Darkmoon Card: Wrath is obtained by gathering all eight of the Storms cards, creating the Storms Deck by right-clicking on any of the cards, and turning in the deck to the Darkmoon Faire. All the individual cards and the deck itself are BoE items. Cards 2 through 4 are world drops, 5 through 8 are world drops from elite mobs, and the Ace drops from end bosses in instances.

Effects on Critical Strike Chance (general)
Using the following java program we're able to simulate what happens to our crit rating when we use this trinket:

public static void main(String[] args) { double crit=0; double x=0; double sum=0; int i=0; Random generator = new Random; for (crit=10;crit<40;crit++){ sum=0; for (i=1;i<1000000;i++){ if (generator.nextInt(100)+1>crit+x) x+=0.72; else x=0; if (x>0.72*20) x=0.72*20; sum+=x; }       System.out.println("  "+crit+""+(crit+sum/i)+" "+((double)sum/i)*23.6+" "); }       }

This simulates the behavior of the trinket after 1,000,000 strikes. The output is the following:

With those results you should able to determine, considering your base chance to crit, if this trinket is a good choice for your character.

Effects on Critical Strike Chance for Casters
The crit benefits of this trinket are affected by the chance of the player to have a streak of non-crit hits. These streaks are less likely for players who already have a high crit chance. Also, the trinket itself, by increasing crit chance, progressively makes long non-crit streaks less likely.

Streaks of Non-Crits
In this example, resists will not be counted. A player with a 25% crit rate will be assumed to have a 75% non-crit hit rate. The probability that a series of non-crits of a certain length (called a streak on this page) occurring can be calculated by multiplying the non-crit hit rate to the power of its streak length. To clarify; the streak is quantified by presuming event X+1 is a critical. In other words:


 * chance of streak = xy   (where x = non-crit chance and y = length of streak)

So with a 25% crit rate, the probability of having a streak of 10 non-crits is 5.63%. Although each cast has a 75% of being non-crit, the percentages have a cumulative effect when looking specifically for a block (or streak) of 10 non-crits in a row.

However, with the trinket, x in the above formula changes with each new non-crit in the streak, because the buff increases the crit rate cummulatively as the streak continues. The modified version this:

The new Chance of Non-Crit Streak is equal to new Non-Crit Chance times the previous Chance of Non-Crit Streak. The blue * yellow = green numbers demonstrate this formula.


 * 0.7269 * 0.4090 = 0.2973


 * 0.6808 * 0.0513 = 0.0349

As you can see, players with a 25% crit rate will have a crit rate of 32% if they have had 10 non-crits in a row, but the chance of having 10 non-crits in a row is only 3.5%. To get an idea of how effective the trinket is, the number of extra crits contributed by the trinket for each scenario must be calculated and then added together.

In the table above, we listed the chance of having a streak of at least X non-crits in a row. The table below breaks down the contribution of the trinket for each streak scenario. This contribution is calculated by multiplying the bonus critical of the trinket by the incidence of the streak. The incidence can be calculated from the above table by deducting the non-streak chance of X from the chance of X-1. For example, there is a 0.75 incidence of a non-critical, and the incidence of a second non-critical is 0.5567 (these values are fractions of the total number of casts done.) The fraction of streaks that end at 1 (i.e. the second cast is a crit) is 0.1933, since the remainder (0.5567) had a second non-crit event. The final column is the product of this number and the increase in crit caused by the trinket, which tells us the fraction of streaks that ended because of the trinket.

For example, a streak of 5 non-crits followed by one crit has a fraction of 0.0617 of all streaks, and the increase in crit from the Wrath trinket is 3.85%. In other words, Wrath provided a 3.85% increased chance to crit for 6.17% of crit events. The table above shows the first 11 (and most likely) occurrences: that you have a streak of non-crits of length zero, a streak of length 1, a streak of length 2, etc. Adding their fractions together, these 11 possibilities summarises 97.65% of all cases (the remaining cases are streaks longer than 10). If we then add up the fractions of crits caused by the trinket, the wrath trinket increases the incidence of crits by 1.85% of all casts. The real figure is higher than this, but this is pretty close since streaks above 10 are unlikely, and its contribution soon becomes insignificant (if you go to 20, you cover 99.977% of the streaks and an increase of 2.0857%; at 29 it is 99.9998% of cases and an increase of 2.0892%).

The table above was calculated for someone who had a base crit rate of 25% (i.e. excluding contributions by the trinket.) If we convert the summed incidence of crits caused by the trinket and convert it to an equivalent critical strike rating, we get the approximate values below (to 20 iterations).

As your base crit rate increases, the value of this trinket decreases, but it is a very valuable trinket at lower rates. For those who are comparing various trinkets, this trinket has a critical rating of exactly 40 (the equivalent of, without the very useful temporary health bonus, or without the potential increase in spell damage) at a base critical rate of 28.13%. It also has a critical rating of exactly 32 (like the spell critical bonus of, for example) at a base critical rate of 33.32%.

The sections below will be of interest to players concerned about long streaks of non-crits (perhaps warlocks trying to keep up Improved Shadowbolt procs).

Rating the effect of the trinket
The significant effect of the trinket is in how it decreases the Chance of Non-Crit Streak. You can think of Chance of Non-Crit Streak as the opposite of crit chance over a given number of casts. For example, in the above chart, there is a 10.71% chance for a streak of 7 non-crits. That also means that there is a 89.29% chance that the streak will fail, that any one cast out of a block of 7 will crit. However, without the effect of the trinket, the streak chance is 13.35% for 7 casts. Without the trinket, there is an 86.65% chance the streak will fail.

So the trinket has increased the chance of at least one crit in any seven casts by 2.64%. Even though the 7 charges would add 5.38% crit chance to the next cast, the unlikelihood of getting 7 non-crits in a row balances that out. This includes the cumulative effect of the base crit rate and the trinket's increasing charges.

In other words, the effect of the trinket can be expressed in terms of the difference between the likelihood of non-crit streaks with and without the trinket. The following graph shows net crit chance increase caused by the trinket over a number of casts for players with different amounts of base crit.



So, as an example from the graph, a player with a base crit rate of 15% will be about 6% less likely to have a string of 14 non-crits due to the trinket. That is to say, he will be 6% more likely to have any one cast crit out of a block of 14.

As you can see, the more base crit you have, the less benefit you get from the trinket, because the likelihood of long strings of non-crits is so low. Also, as you consider longer and longer streaks, the net benefit of the trinket actually begins to decrease. This is because the chance of one crit in say 18 or 20 casts is so high that any amount of crit increase is insignificant.

Summary
Since Darkmoon Card: Wrath increases a player's crit rate only while the player is not critting, it is reducing its chance to build charges while building charges. So even though at 7 charges the trinket increases crit rate by 5.38%, we have to balance that with the increasing unlikelihood that the player will have 7 non-crits in a row, an unlikelihood affected by base crit plus added crit from the trinket.

Instead of looking flatly at crit increase, we can look at the difference in likelihood of non-crit streaks between streaks affected by the trinket and streaks with only base crit. The difference in these is the increased chance that any one cast across the streak will crit. Interestingly, we found that the benefit of the trinket maxes out at a certain point, and that longer streaks of non-crits are so unlikely that the trinket adds little to the cumulative effect of the base crit rate. Using this difference as a yardstick for the trinket's effect, here are some results for various base crit rates:


 * Average Effect is the crit rate increase for streaks that have a ~50% chance of occurrence
 * Maximum Effect is the largest difference in likelihood of non-crit streaks caused by the trinket

Simulation
A simple script that runs simulations of stacking Wrath buffs for casters, and computes the effective crit increase, can be found here.