Elitist Jerks A Mask of All Occasions, expected completion time.

Rate this Entry

# A Mask of All Occasions, expected completion time.

Posted 10/20/10 at 5:56 PM by Hamlet
This is in honor of my guildmate Vosk ( The World of Warcraft Armory - Vosk @ Mal'Ganis - Achievements ), who recently completed the achievement A Mask for All Occasions, becoming one of a handful of people worldwide to complete all of the achievements in WoW. The luck involved in this achievement prompts a lot of discussion, particularly in Vosk's case, since he would have been the first player in the world to complete all achievements if he'd been luckier with masks last year. It's not too difficult to compute how long one can expect to take to complete this achievement in the average case, but many people get tripped up on the questions of probability involved.

By far the simplest way to find the expected time to compute all 20 masks is this: if you have N masks already, then when you get a new mask, the probability that it will be a new one is (20-N)/20. So if the overall chance of getting a mask on a given Trick or Treat attempt is P, then on a given loot, the chance of a new mask is P*(20-N)/20. Inverting, the expected number of loots to obtain each new mask is 20/(P*(20-N)). Simply add these up to find the total.

$\Sigma_{n=0}^{19}{\frac{20}{P(20-n)}} = \frac{20}{P}\Sigma_{n=1}^{20}{\frac{1}{n}}$

The summation is the 20th harmonic number (Harmonic number - Wikipedia, the free encyclopedia ), which evaluates to 3.60. Therefore the final expected number of attempts is

$\frac{72}{P}$.

If we set P=1, to consider only attempts that result in masks, we see that you can expect to loot 72 masks in order to complete the achievement. So those of you complaining about duplicate masks should consider that an average result is 52 repeats before completing the achievement.

Another useful exercise is to know just far along you really are, based on the number of masks you have. This can be quite deceptive. We simply look at the intermediate sums from the above equation. For example, carrying the sum to only n=9 produces a value of 13.4 (still assuming P=1). So when you collect your 10th mask, you are only 13.4/72 = 18.6% of the way there. When you collect your 15th mask, you are only 36.5% done. To finally be more than halfway toward the end of the achievement, you need to have looted 18 masks.

Actually estimating the number of total attempts requires knowing P. Good data here is hard to find, since Wowhead data does include Trick outcomes, and the number of masks changed between years. But a good estimate seems to be P=0.2 (based on Wowhead and a few other sources, and asking Vosk), which would result in about 360 attempts total. Given that Hallow's End lasts less than two weeks, and there are only 168 hours in a week, this can be finished in two years with extreme diligence and above average luck, or in three years with good diligence.

Sadly, luck was not with Vosk, who was more than extremely diligent, sometimes waking up every hour to loot another bag, and still had well over 100 masks before he finally got a first Male Orc.
Posted in Math