Scaling Type:
Hyperbolic scaling
Detailed Explanation
The official description says +10% chance to open chests for free. But the real in-game chance is actually closer to 9.1% when you have 1 key.
The reason it says 10% chance in-game is because that is the raw input number used for the hyperbolic scaling formula.
If you calculate the hyperbolic scaling formula for 1 key you get 0.0909 => ~9.09% chance with 1 key (rounded up to 9.1% in the graph below)
The chests that have a chance to become free are the ones you pay gold for. If you successfully open a chest for free it will show after opening a brown chest:
The reason it says 10% chance in-game is because that is the raw input number used for the hyperbolic scaling formula.
If you calculate the hyperbolic scaling formula for 1 key you get 0.0909 => ~9.09% chance with 1 key (rounded up to 9.1% in the graph below)
The chests that have a chance to become free are the ones you pay gold for. If you successfully open a chest for free it will show after opening a brown chest:
How it Stacks
You can read about how hyperbolic scaling works in general here. hyperbolic scaling. If you're just interested in the key, here's how it works for the key specifically
if n = number of keys, then this formula can be used to describe the chance of getting a free chests from X amount of keys
If you plug this out on a chart, you can see this type of scaling:
You can never reach a 100% chance no matter how many keys you have. You can get very close, but it will require an absurd amount of keys.
With 100 keys you have a ~90.9% chance
With 500 keys you have a ~98.03%
With 1000 keys you have a ~99.01%
To get a 99.99% chance you need 99 990 keys
Even with 1 million keys you don't have a 100% chance but a ~99.999% chance
if n = number of keys, then this formula can be used to describe the chance of getting a free chests from X amount of keys
$$
\text{chance}(n)
= \frac{\text{chance}_\text{raw}(n)}{\text{chance}_\text{raw}(n) + 1}
$$
If you plug this out on a chart, you can see this type of scaling:
You can never reach a 100% chance no matter how many keys you have. You can get very close, but it will require an absurd amount of keys.
With 100 keys you have a ~90.9% chance
With 500 keys you have a ~98.03%
With 1000 keys you have a ~99.01%
To get a 99.99% chance you need 99 990 keys
Even with 1 million keys you don't have a 100% chance but a ~99.999% chance