# Drop rate

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Drop Rate is the probability that a monster is expected to yield a certain item when killed once by a player. When calculating a drop rate, divide the number of times you have received the certain item, by the total number of that NPC that you have killed. For example:

A common misconception is that you are guaranteed that item when you kill the NPC $x$ number of times, where $\frac{1}{y}$ is the drop rate. You are never guaranteed anything, no matter how many times you kill that monster. The drop rate is simply the probability of getting a certain drop in one kill. The probability that a monster will drop the item at least once in $x$ kills is 1 minus the probability that it will not drop that item in $x$ kills, or $1 - \left(1 - \frac{1}{y}\right)^x$, where x= number of kills, and y= drop rate.

For example, if dust devils are expected to drop a Dragon chainbody once out of 15000 kills, then the probability that a player will get at least one Dragon chainbody after 15000 kills is

$1-\left(\frac{14999}{15000}\right)^{15000}$

Which is approximately 63.21%. Similarly, we can solve for the number of Dust Devils you need to kill to have a 90% probability of getting one when you kill them:

$1-\left(\frac{14999}{15000}\right)^{x} > 0.9$

$\left(\frac{14999}{15000}\right)^{x} < 0.1$

Which yields the answer $x>34538$. There is also an equation for computing the probability of a certain amount r of a particular drop after n amount of kills:

$P(r,n)={}^{n}\textrm{C}_{r} p^{r}q^{n-r}$
And if you take the sum of this equation from when r=1 until r=n you get the probability of at least 1 drop of a particular item after n kills:
$\sum_{r=1}^{n}{}^{n}\textrm{C}_{r} p^{r}q^{n-r}=1-\left (1-\frac{1}{n} \right )^{n}$

## Estimation

Drop rates are often quite difficult to obtain, as an accurate estimation of one requires thousands of kills. Because of this, some players who wish to calculate drop rates keep a list of items that a monster drops after each kill, sometimes called a "drop log." Then they calculate the percentage by dividing the number of desired drops by the total number of kills. All monsters found on this Wiki contain a list of the items they drop. Behind those items you will often find between brackets a drop rate indication for that item. The drop rate of items has been divided into five different groups displayed below.

Rarity Drop rate^(-1) Example*
Always 1 Bones
Common 2-50
Uncommon 51-100 Rune armour
Rare 101-512 Abyssal whip
Very rare 513+ Draconic visage

* Examples are only given as indication because they depend on the monster that drops it. An item dropped by a boss monster could be a common item while it would be very rare for normal monsters.

## Trivia

If we let x be an arbitrary number and $1/x$ be the drop rate for a particular drop, the larger x gets (in other words, the rarer the drop is), the closer the probability of obtaining that item in x kills approaches $1 - \frac{1}{e}$, or approximately $0.63212$, where e is the exponential constant $\approx{2.718281828459045}$. We can express this limit as follows:

$\lim_{x \to \infty} 1 - \left(1 - \frac 1x\right)^x = 1 - \frac 1e$

This follows from the definition of $e$:

$e = \lim_{n \to \infty} \left(1 + \frac 1n\right)^n
=\sum_{i=0}^{\infty} \frac{1}{i!}$

This leads to the conclusion that, given a drop rate of $\frac{1}{r}$, the approximate chance of not receiving a drop after $n$ kills is $\left(\frac{1}{e}\right)^\frac{n}{r}$. Note that this is only accurate for large values $r$.