Drop rates
What is the chance of dropping a reward?
On many of the guides you will see the drop-rate for a particular reward given. This will typically be expressed as a fraction or a percentage. For example, the rate of drop-rate for the 
Chapeau Noir in 
Thieving is 1/20,000 (0.005%).
The drop-rate of the reward shows you probability of the item dropping, but it is not the same as the chance of the reward being given. For example, if you were to perform this action 20,000 times you might reasonably expect to get this item, however, there is only a 63% chance of this happening.
For example, the chances for the Chapeau Noir to drop at least once for a given number of attempts can be seen on this table.
| Chance of success | Number of attempts |
|---|---|
| 50% | 13,863 |
| 90% | 46,051 |
| 99% | 92,102 |
It might be surprising to learn that farming for rare items is often going to take considerably longer than you may have initially thought based on the drop-rate. Of course, you might get lucky and drop this reward on your first attempt!
Visual representation of drop rates
To visualize what this means, here is a graph showing four different drop rates:
| Drop rate | Color | #Attempts for 80% Chance |
|---|---|---|
| 1/10,000 | Red | 16,000 |
| 1/20,000 | Orange | 32,000 |
| 1/30,000 | Blue | 48,000 |
| 1/40,000 | Green | 64,500 |
So keep grinding, and see the next section for some of the math behind the probabilities of drops.
Calculating the chance of drops
The probability (P) for a given drop-rate (k) over a fixed number of attempts (n) is given by:
[math]\displaystyle{ P = 1 - \left ( 1 - \dfrac{1}{k} \right ) ^ {n} }[/math]
If the drop-rate is low (which is typical for rare drops) then you can approximate this as:
[math]\displaystyle{ P \approx 1 - \left ( \dfrac{1}{2.718} \right ) ^ {\left ( \dfrac{n}{k} \right )} }[/math]
For example, for the Chapeau Noir there is a 1 in 20,000 (k) chance of this item dropping, and 100 attempts gives a probability of:
[math]\displaystyle{ P = 1 - \left ( 1 - \dfrac{1}{20,000} \right ) ^ {100} = 0.005 = 0.5 \% }[/math]
This means that you have a 0.5% chance of discovering at least one of these after 100 attempts.
Calculating the number of attempts
The number of attempts (n) for a given drop-rate (k) with a certain probability (P) is given by:
[math]\displaystyle{ n = \left \lceil \dfrac {ln(1 - P)} {\ln \left (\dfrac{(k - 1)}{k} \right )} \right \rceil }[/math]
For those unfamiliar with math notation, the outer brackets just mean take the largest whole number.
If the drop-rate is low which is typical for rare drops then you can approximate this as:
[math]\displaystyle{ n \approx \left \lceil -k \cdot \ln(1 - P) \right \rceil }[/math]
For example, for the Chapeau Noir there is a 1 in 20,000 (k) chance of this item dropping, and we want a 75% chance of getting at least one, then we would need:
[math]\displaystyle{ n \approx \left \lceil -20,000 \cdot \ln(1 - 0.75) \right \rceil = 27,726 }[/math]
This means that you would need 27,726 attempts to give you a 75% chance of discovering that item.
Further reading
Further information can be found:
- Online calculators that you can use to calculate drop-rates.
- Information on the Binomial Distribution.
