* copy/ paste from Reddit;
Uber Unique Math - The struggle explained
Hello fellow D4 Redditors! I recently responded to a post about uber uniques and I figured I could expand on that response and make a full thread explaining the math behind uber unique farming.
TW: Math
TLDR: On average, 1250 kills or 1800 kills, depending on class, to find the entire uber unique drop pool for your class. Full explanation below.
**Introduction**
My name is Matt and I have a master's degree in mathematical sciences from the University of West Florida, with an emphasis on statistics and data analysis. What I'll be doing here is not talking specifically about the drop chance of an uber unique, but the number of Duriel kills you should expect in order to find one of each for your class. I'll be doing this using a couple different probability distributions, specifically the Binomial Distribution, the Negative Binomial Distribution, and the Geometric distribution. I hope to put the uber unique grind into perspective and provide some explanation on why some people are struggling even after hundreds of kills.
**Distributions**
* **Binomial**
* The main characteristic of the binomial distribution is that each trial (in this case a kill of Duriel) has two possible outcomes, success (uber unique) and failure (no uber unique).
* The probability of success for these trials is denoted by p and it doesn't change across trials.
* The trials are independent, so the outcome of one trial doesn't impact the probability of another trials outcome.
* There is a fixed number of trials, denoted by n.
* **Geometric**
* Say you are performing binomial trials and you want to know the probability of getting a certain number of failures before you have your first success. That's what the geometric distribution does.
* The mean of the Geometric distribution is simply 1/p and it gives us the **average number of failures before one success**. This is what I'm most interested in here.
* The geometric distribution is a special case of the third distribution we need to know about, the negative binomial distribution.
* **Negative Binomial**
* The negative binomial distribution is very similar to the geometric, but you aren't limited to 1 success. You are able to choose the number of successes. This is the main distribution I'll be using for this because it will let us calculate the average number of Duriel kills before 5 or 6 successes (more on that in a sec).
**Calculations**
There are a few more pieces of information that we need before we start the real math. First, not all uber uniques drop for each class. Sorcerer, Druid, and Rogue each can drop 5 ubers, while Barbarian and Necromancer can drop 6. I'm going to abbreviate these two groups as SDR group and BN group.
The apparent uber unique drop rate from Duriel is 2% so that's something we will need.
Ok time for math. The thread I responded to earlier was from someone who had 6 uber uniques drop total, but he had 4 of one and 2 of the other. They were understandably frustrated, but this isn't unusual or unlucky, it's pretty expected. To show that, we can use the negative binomial distribution mean. The SDR group has 5 possible uber drops, so the probability for one specific one to drop is p=1/5. If you want the average number of uber uniques that have to drop for you to get all 6, the formula for that is mean = r/p, where r is the number of successes and p is the probability of success. So for the SDR group you get r/p = 5/(1/5) = 25. That means, on average, it would take **25 uber uninque drops** to get 1 of each. For the BN group it's even worse: you end up with r/p = 6/(1/6) = 36 uber unique drops. Having only 6 drops it was no wonder this person didn't get very many different ones.
Now I went a step further and calculated the range of "usual" values for how many uber uniques you need to drop before getting all of them. In statistics, "usual" values are usually defined as anything within two standard deviations of the mean. So, the formula for the standard deviation of the negative binomial distribution would be s=sqrt(r\*(1-p))/p. Using that formula with p=1/5 or 1/6, and r as 5 or 6, you get the standard deviation for SDR group is 4.5. The standard deviation for BN group is 5.5. This means that these are the "usual" numbers of unique drops you'd need to get one of each:
**SDR Group** 16 to 34 uber drops
**BN Group** 25 to 47 uber drops
That means it can take almost 50 uber unique drops, depending on class, before you get all of them and that's still **completely normal**. Not great news.
This leads me into the even worse news... the number of Duriel kills this translates to. Using the same type of calculations, we can find the same values for Duriel kills, we just have to factor in the 2% drop rate per unique drop. I'll quickly summarize the results from that here:
**SDR Group** Average kills = 1250, usual number of kills 1179 to 1321
**BN Group** Average kills = 1800, usual number of kills 1715 to 1885
This means that even on the low end, if you aren't above **1000 Duriel kills** you are not very likely to collect every uber in your classes loot pool. The barb and necro group needs to get close to 2000 before it's unusual.
**Conclusion**
To quickly summarize the results here: Depending on class, you need 1250 or 1800 Duriel runs to have an "average" chance to get every uber unique. Not only is this depressing, it's a huge amount of time and resources to sink into this endeavor.
Well, I hope this post helps at least one person better understand the uber unique grind and the math behind why you may not be finding your favorite gear right away. If you've read this far, you're a nerd (like me) and I hope you enjoyed it. If any other math people spot a mistake just lemme know and I'll do my best to correct it. Hopefully this stuff can encourage more conversation on the topic.
Thanks for reading.