Darkmoon Card: Wrath

• Darkmoon Card: Wrath
• Binds when picked up
• Unique
• Trinket
• Requires Level 70
• Item Level 100Disenchants into:
   [Void Crystal]: 1-2
• Equip: Template:Spelllink
• Sell Price: 10 0 0

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 the set of eight Storms cards, creating the  [Storms Deck] by clicking on the Ace of Storms, 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 heroic instances.

Effects on Critcial 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 chance of a non-crit hit streak of a certain length to occur can be found by carrying the non-crit chance to the power of the length of the streak. In other words:

• chance of streak = x^y    (where x = non-crit chance and y = length of streak)

Streaks without Trinket

# of Casts	Crit Chance	Non-Crit Chance	Chance of Non-Crit Streak
1	0.2500	0.7500	0.7500
2	0.2500	0.7500	0.5625
3	0.2500	0.7500	0.4219
4	0.2500	0.7500	0.3164
5	0.2500	0.7500	0.2373
6	0.2500	0.7500	0.1780
7	0.2500	0.7500	0.1335
8	0.2500	0.7500	0.1001
9	0.2500	0.7500	0.0751
10	0.2500	0.7500	0.0563

So with a 25% crit rate, the chance 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 progressively increases the crit rate. So it works like this:

Streaks with Trinket

# of Casts	Crit Chance	Non-Crit Chance	Chance of Non-Crit Streak
1	0.2500	0.7500	0.7500
2	0.2577	0.7423	0.5567
3	0.2654	0.7346	0.4090
4	0.2731	0.7269	0.2973
5	0.2808	0.7192	0.2138
6	0.2885	0.7115	0.1521
7	0.2962	0.7038	0.1071
8	0.3038	0.6962	0.0745
9	0.3115	0.6885	0.0513
10	0.3192	0.6808	0.0349

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 can raise their crit rate to 32% if they have 10 non-crits in a row, but the chance of having 10 non-crits in a row is only 3.5% Its tempting to say 7% crit rate increase occurs 3.5% of the time, yielding 0.242% overall crit increase. However, with a 3 non-crit streak, crit rate is increased by 1.5%. There is a 40% chance for a 3 non-crit streak. So 1.5% increase 40% of the time yields 0.6% overall increase. Because the effect is cumulative over any streak, the expectation must be taken.

The expected value of a random quantity is the probability of it occuring times it's value. We can use this principle to compute the value of the trinket. In the table above, we listed the chance of having a streak of at least X non-crits in a row. The chance of having exactly a streak of length X is this number times the probability of then having a crit on the X+1 cast. This is listed below. The expected crit contribution is listed in the far right column.

Streaks of Length X

X = Streak Length	Additional Crit from Trinket	Streak Chance	Expected Value
0	0.0000	0.2500	0.0000
1	0.0077	0.1933	0.0015
2	0.0154	0.1477	0.0023
3	0.0231	0.1117	0.0026
4	0.0308	0.0835	0.0026
5	0.0385	0.0617	0.0024
6	0.0462	0.0451	0.0021
7	0.0538	0.0325	0.0017
8	0.0615	0.0232	0.0014
9	0.0692	0.0164	0.0011
10	0.0769	0.0114	0.0009
TOTALS:	-	0.9765	0.0185

For example, if there is a streak of 5 non-crits followed by one crit, this streak has a 0.0617 chance of occuring, then when the crit occurs the Wrath trinket is providing 3.85% additional crit chance. This means 6.17% of the time, Wrath provides 3.85% increased chance to crit. The table above shows the 11 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. Together, these 11 possibilities cover 97.65% of the cases (the remaining cases are streaks longer than 10). Thus if we sum the expected contributions of the wrath card over these occurrences, we see that 97% of the time, wrath provides 1.85% increases chance to crit. To get a better estimate, we can extend the tables above to streaks of length 20, or 30, but since these streaks become very unlikely, it becomes less important to consider them.

The table above was done for a base crit rate of 25%, not including the trinket. Below we show the expected crit rating of the trinket for other base crit rates.

Expected Crit Rating

Base Crit Rate	Expected Wrath Crit Rating
45%	20
40%	24
35%	30
30%	37
25%	46
20%	58
15%	76
10%	99
5%	129

We see that as your base crit rate increases, the value of this trinket decreases. However for low crit rates, it is a very valuable trinket indeed. For players concerned with their average crit rate, and how this trinket effects it, this is the answer. 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:

Net Benefit from Trinket for Different Base Crit Rates

Base Crit	Average Effect	Maximum Effect
10%	4.9% crit increase 54% of the time	12.7% crit increase 10.2% of the time
15%	2.8% crit increase 49% of the time	6.7% crit increase 10% of the time
20%	1.5% crit increase 50% of the time	4% crit increase 9.4% of the time
25%	0.6% crit increase 56% of the time	2.6% crit increase 10.7% of the time
30%	0.5% crit increase 49% of the time	1.8% crit increase 10% of the time

• 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

Raw Data

Net Crit Increase from Trinket

	Chance of Non-Crit Streak (10% base)		Crit Chance Increase		Chance of Non-Crit Streak (15% base)		Crit Chance Increase		Chance of Non-Crit Streak (20% base)		Crit Chance Increase
# of casts	w/o trinket	with trinket			w/o trinket	with trinket			w/o trinket	with trinket
1	0.9000	0.9000	0.0000		0.8500	0.8500	0.0000		0.8000	0.8000	0.0000
2	0.8100	0.8031	0.0069		0.7225	0.7160	0.0065		0.6400	0.6338	0.0062
3	0.7290	0.7104	0.0186		0.6141	0.5976	0.0166		0.5120	0.4973	0.0147
4	0.6561	0.6230	0.0331		0.5220	0.4941	0.0279		0.4096	0.3864	0.0232
5	0.5905	0.5415	0.0490		0.4437	0.4048	0.0389		0.3277	0.2972	0.0305
6	0.5314	0.4665	0.0649		0.3771	0.3285	0.0486		0.2621	0.2263	0.0358
7	0.4783	0.3983	0.0799		0.3206	0.2641	0.0565		0.2097	0.1706	0.0391
8	0.4305	0.3371	0.0934		0.2725	0.2102	0.0622		0.1678	0.1273	0.0405
9	0.3874	0.2826	0.1048		0.2316	0.1658	0.0658		0.1342	0.0940	0.0402
10	0.3487	0.2348	0.1139		0.1969	0.1294	0.0674		0.1074	0.0687	0.0387
11	0.3138	0.1932	0.1206		0.1673	0.1001	0.0673		0.0859	0.0497	0.0362
12	0.2824	0.1576	0.1249		0.1422	0.0766	0.0657		0.0687	0.0355	0.0332
13	0.2542	0.1273	0.1269		0.1209	0.0580	0.0629		0.0550	0.0252	0.0298
14	0.2288	0.1018	0.1270		0.1028	0.0435	0.0592		0.0440	0.0176	0.0264
15	0.2059	0.0807	0.1252		0.0874	0.0323	0.0550		0.0352	0.0122	0.0230
16	0.1853	0.0633	0.1220		0.0743	0.0237	0.0505		0.0281	0.0083	0.0198
17	0.1668	0.0492	0.1176		0.0631	0.0173	0.0459		0.0225	0.0056	0.0169
18	0.1501	0.0378	0.1123		0.0536	0.0124	0.0412		0.0180	0.0038	0.0142
19	0.1351	0.0288	0.1063		0.0456	0.0088	0.0368		0.0144	0.0025	0.0119
20	0.1216	0.0217	0.0999		0.0388	0.0062	0.0325		0.0115	0.0016	0.0099

 

Net Crit Increase from Trinket (2)

	Chance of Non-Crit Streak (25% base)		Crit Chance Increase		Chance of Non-Crit Streak (30% base)		Crit Chance Increase
# of casts	w/o trinket	with trinket			w/o trinket	with trinket
1	0.7500	0.7500	0.0000		0.7000	0.7000	0.0000
2	0.5625	0.5567	0.0058		0.4900	0.4846	0.0054
3	0.4219	0.4090	0.0129		0.3430	0.3318	0.0112
4	0.3164	0.2973	0.0191		0.2401	0.2246	0.0155
5	0.2373	0.2138	0.0235		0.1681	0.1503	0.0178
6	0.1780	0.1521	0.0258		0.1176	0.0994	0.0182
7	0.1335	0.1071	0.0264		0.0824	0.0650	0.0173
8	0.1001	0.0745	0.0256		0.0576	0.0420	0.0156
9	0.0751	0.0513	0.0238		0.0404	0.0268	0.0135
10	0.0563	0.0349	0.0214		0.0282	0.0169	0.0113
11	0.0422	0.0235	0.0187		0.0198	0.0105	0.0092
12	0.0317	0.0156	0.0160		0.0138	0.0065	0.0074
13	0.0238	0.0103	0.0135		0.0097	0.0039	0.0057
14	0.0178	0.0067	0.0111		0.0068	0.0024	0.0044
15	0.0134	0.0043	0.0091		0.0047	0.0014	0.0033
16	0.0100	0.0027	0.0073		0.0033	0.0008	0.0025
17	0.0075	0.0017	0.0058		0.0023	0.0005	0.0019
18	0.0056	0.0011	0.0046		0.0016	0.0003	0.0014
19	0.0042	0.0006	0.0036		0.0011	0.0002	0.0010
20	0.0032	0.0004	0.0028		0.0008	0.0001	0.0007

Simulation

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

Damage

At only a 10% base crit rate, casting a 2.5 second spell as fast as possible allows only 4 spells to stack giving a total increase of 5.86% crit chance. If you assume 1.5k normal damage and 3k crit damage that's an average boost of 87.9 damage over the four casts. That comes to a boost of 21.975 damage per spell cast. If an instant cast spell is assumed such that you only have the global cooldown of 1.5 seconds it allows up to 6 stacks of the buff giving a 17.25% chance to crit. This is a boost of 258.75 damage over 6 attacks, giving an average bonus of 43.125 damage per attack in the best case assuming 1.5k normal damage and talent to make critical hits double damage. This makes the [Glowing Crystal Insignia] superior. The calculation listed on the site is actually extremely conservative considering that in most fights it's over before 2 minutes is up.

External links

• Wowhead
• WoWDB