Key
Key Megabonk – Explained
The Key's 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.
This is because the key uses hyperbolic scaling and does not scale linearly. If it scaled linearly you would have a 100% chance to open a free chest with 10 copies but that is NOT the case as it would be way too overpowered.
Instead the 10% you see in the description is used in a hyperbolic scaling formula as the raw input number. But when it has been run through the formula and done all the math, it comes out to approximately 9.1% with 1 key. And if you have 10 keys it would be a 50% chance. You can see more data points in the graph under "How it Stacks"
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:
This is because the key uses hyperbolic scaling and does not scale linearly. If it scaled linearly you would have a 100% chance to open a free chest with 10 copies but that is NOT the case as it would be way too overpowered.
Instead the 10% you see in the description is used in a hyperbolic scaling formula as the raw input number. But when it has been run through the formula and done all the math, it comes out to approximately 9.1% with 1 key. And if you have 10 keys it would be a 50% chance. You can see more data points in the graph under "How it Stacks"
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
As mentioned, the key scales using a hyperbolic scaling formula. This means every key will contribute in going towards a 100% chance but each key will have diminishing returns. You will never reach a 100% chance to open a free chest regardless of how many keys you have.
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:
Example table:
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:
Example table:
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 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
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