imported>Code2004: Added section on calculating confidence intervals for drop rates according to normal approximation intervals as defined http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval

2012-09-11T10:19:53Z

Added section on calculating confidence intervals for drop rates according to normal approximation intervals as defined http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval

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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:

*[[Bones]] have a 100% drop rate from [[Chicken]]s
*[[Feathers]] have a 75% drop rate from Chickens

A common misconception is that you are guaranteed that item when you kill the NPC <math>x</math> number of times, where <math>\frac{1}{x}</math> 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 <math>x</math> kills is 1 minus the probability that it will '''not''' drop that item in <math>x</math> kills, or <math>1 - \left(1 - \frac{1}{y}\right)^x</math>, where x= number of kills, and y= drop rate.

For example, if [[dust devil]]s 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

<math>1-\left(\frac{14999}{15000}\right)^{15000}</math>

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:

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

<math>\left(\frac{14999}{15000}\right)^{x} < 0.1</math>

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

 <math>P(r,n)={}^{n}\textrm{C}_{r} p^{r}q^{n-r}</math> 
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:
 <math>\sum_{r=1}^{n}{}^{n}\textrm{C}_{r} p^{r}q^{n-r}=1-\left (1-\frac{1}{n} \right )^{n}</math> 
== 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.

{|class="wikitable" width="400"
!Rarity
!Drop rate^(-1)
!Example*
|-
!Always
|1
|[[Bones]]
|-
!Common
|2-50
|
[[Coins]]
|-
!Uncommon
|51-100
|[[Rune armour]]
|-
!Rare
|101-512
|[[Abyssal whip]]
|-
!Very rare
|513+
|[[Draconic visage]]
|}

<nowiki>*</nowiki> 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.

===Confidence Intervals===
''This section should only be considered by people who understand algebraic manipulation and have a basic understanding of a statistical model.''

It is given to us that the confidence interval for the success probability of a model <math>X\sim B(n,p)</math> may be expressed as the formula<ref>Mayfield, Philip. [http://www.sigmazone.com/binomial_confidence_interval.htm ''Understanding Binomial Confidence Intervals'']</ref>:

<math>C=p\pm z_{1-\frac{\alpha}{2}}\sqrt{\frac{p(1-p)}{n}}</math>

''Where:''
* <math>p</math> - the assumed probability of success given by the ratio of successes to sample size. To clarify: if one were to gain 2 [[Divine Sigil]]s after 2000 [[Corporeal Beast|Corp]] kills, the assumed probability of success would be <math>\frac{2}{2000}=\frac{1}{1000}=0.001</math>
* <math>z_{1-\frac{\alpha}{2}}</math> - this is the critical standard score such that <math>P(Z\leq z_{1-\frac{\alpha}{2}})\approx1-\frac{\alpha}{2}</math> for <math>Z\sim N(0,1)</math>. This z-value may be found by checking with [http://en.wikipedia.org/wiki/Standard_normal_table#Cumulative_table this table]. Information on how to read this table may be found [http://en.wikipedia.org/wiki/Standard_normal_table#Reading_a_Z_table here].
* <math>\alpha</math> - the confidence error you wish your interval to represent. An example value may be 0.05 (this represents 95% confidence).
* <math>n</math> - the amount of trials you've conducted. In the example used in the definition of 'p', this value would be 2000.

To save the reader time, a list of possible z-values is supplied:
{| class="wikitable" width="300"
|-
! <math>\alpha</math> !! Confidence level !! <math>z_{1-\frac{\alpha}{2}}</math>
|-
| 0.2 || 80% || 1.28
|-
| 0.1 || 90% || 1.64
|-
| 0.05 || 95% || 1.96
|-
| 0.01 || 99% || 2.57
|}

;Example of usage:
Consider the following case: we have killed a combined total of 500 [[Black Dragon]]s and have gained 10 [[Draconic Visage]]s between us. This suggests that we take <math>p=\frac{10}{500}=0.02</math> and <math>n=500</math>. Now let us say that we wish to create a 95% confidence interval for our p-value (this is to say that <math>\alpha=0.05</math> and <math>z_{1-\frac{\alpha}{2}}=1.96</math>). Our confidence interval is constructed as follows:

<math>C_{lowerbound}=p-z_{1-\frac{\alpha}{2}}\sqrt{\frac{p(1-p)}{n}}=0.02-1.96\sqrt{\frac{0.02(1-0.02)}{500}}=0.00772846\approx\frac{1}{129}</math>

And...

<math>C_{upperbound}=p+z_{1-\frac{\alpha}{2}}\sqrt{\frac{p(1-p)}{n}}=0.02-1.96\sqrt{\frac{0.02(1-0.02)}{500}}=0.0322715\approx\frac{1}{31}</math>

What this means is that we can be about 95% sure that the the drop rate of [[Draconic Visage]]s (from [[Black Dragon]]s) is somewhere between 1 in 31 and 1 in 129.

;Notes on usage:
*This method of calculating confidence intervals relies on being able to approximate our binomial model as a normal distribution -- as such, most statisticians will not use this method unless <math>np>5</math> and <math>n(1-p)>5</math>.<ref>Wikipedia - [http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval ''Binomial proportion confidence interval''].</ref>

== Trivia ==
If we let ''x'' be an arbitrary number and <math>1/x</math> 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 <math>1 - \frac{1}{e}</math>, or approximately <math>0.63212</math>, where e is the exponential constant <math>\approx{2.718281828459045}</math>. We can express this limit as follows:

<math>\lim_{x \to \infty} 1 - \left(1 - \frac 1x\right)^x = 1 - \frac 1e</math>

This follows from the definition of <math>e</math>:

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

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

== Notes ==
{{reflist}}
[[Category:Mechanics]]

Drop rate - Revision history

imported>Code2004: Added section on calculating confidence intervals for drop rates according to normal approximation intervals as defined http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval