In threads about mount drops or other rare items, I often see people answer the question
“How many attempts will I need before I get the item?”
with
“It’s random, so it could take 1 attempt, or it could never drop.”
That answer always annoys me because, while it’s technically true, it’s not helpful, and it can make players believe that getting a drop after 100 attempts is as likely as getting a drop after 1000 attempts. Or equivalently, there are an equal number of players who will get a drop at 100 and 1000 attempts.
In this thread, I want to explain a basic framework that can quantify how lucky or unlucky a player is and give a reasonable estimate for the max number of attempts a player could expect a rare item to take. But first, some background information.
Geometric Distribution:
Mount drops can be modeled using something called a geometric distribution. I’ll try to explain the concepts in a way that is easy to understand, even if you don’t have any background with probability.
Geometric distributions are used to model situations where there is an experiment with two outcomes that are labeled as a success or failure. Each experimental trial’s outcome (success or failure) has a fixed probability that will not change throughout your trials. If you meet these conditions, you can use the geometric distribution to calculate the probability of having your first success occur after n number of trials. Or equivalently failing n-1 times then succeeding.
In our case, the experiment is looting a mob/chest, and the two outcomes are getting the mount or not getting the mount. The drop rate for the item is fixed between each run; invincible will have a 1% drop rate at 1 attempt and at 200 attempts.
A Coin Flip:
One requirement to be able to use the geometric distribution is that the outcome of one trial cannot influence the outcome of another (another way of saying the probability of success/failure does not change). This property is called Independence, and a simple example of an independent event is a coin flip.
As you know, the probability of flipping heads/tails (assuming the coin isn’t rigged) is 50%. Even if you flip heads 10 times in a row, the probability that the next coin flip will be heads is 50%. Your previous flips do not affect the outcome of your next flip.
A neat property of independent events is that you can calculate the overall probability of a sequence of events by multiplying the probability of each event together. So the probability of flipping heads 10 times in a row is (1/2)^10 or 0.0976%.
Using the Geometric Distribution:
The geometric distribution uses the property of independence to calculate the probability of getting your first success after n trials with the following equation.
P(X = N) = (1 - P)^N-1 * P
Where P is the probability of success, 1-P is the probability of failure, and N is the number of trials.
Using a 1% drop rate as an example, the probability of getting a drop at 50 attempts is
P(X = 50) = (1-0.01)^49*0.01 = 0.135%
Cumulative Probability:
In the above example, we found the probability of getting a drop at a specific number of attempts. But what if we want to know the probability of taking N or more attempts? Say 50 or more, for example.
It’s pretty simple you just calculate the probability of getting a drop at 1,2,3…49 attempts, add them all together, then subtract that number from 1. Doing this by hand is a pain, but online calculators can do this for you. If you do the above steps, you’ll find that the probability of requiring 50 or more attempts before getting a 1% drop is 61.11%.
You can use this online calculator to find the cumulative probability of a specified number of attempts. The lower cumulative represents the probability of N or fewer attempts, and the upper cumulative represents N or more attempts
https://keisan.casio.com/exec/system/1180573193
Quantifying Luck: The Law of Large Numbers
Telling someone that for a 1% drop rate, they have a 90% chance to have seen a mount drop by 228 attempts can be confusing. It goes against people’s intuition that the drop rate is constant, so how can some number of attempts be different from another.
I’ve found another way of explaining it using something called the law of large numbers. The law of large numbers states.
“the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed.”
What this means is that instead of talking about the probability of seeing a mount after 228 attempts like the above example, we can say that for a large enough population of mount farmers, 90% of them will have gotten a drop by 228 attempts. Framing it in this way is easier to understand IMO.
The point at which 90% of players would have gotten a drop is also called the 90th percentile. This online calculator can be used to find whatever percentile you are interested in.
https://keisan.casio.com/exec/system/1180573192
You just enter the drop rate and percentile as a decimal.
The percentile a player is a part of indicates how lucky/unlucky they are. If someone is in the 95th percentile, that means 95% of other players would have already gotten a drop, but they still haven’t, so they are unlucky. If they are part of the 5th percentile, only 5% of players would have gotten a drop by then, so they are lucky.
You can also use percentiles to calculate a reasonable maximum number of attempts a random person could expect a farm to take. By assigning a percentile as a ‘reasonable maximum’, we can give a good answer to
“How many attempts will I need before I get the item?”
Setting the maximum number of attempts to the 99th percentile will cover the vast majority of players, so that’s usually the number I tell someone. Of course, it could take more than that many attempts but only for the top 1% of unlucky players.
