There are gems in the game that give you a chance to not use ammo. They effectively increase your clipsize.


Expected Ammo For Varying Clip Sizes and Varying Probabilities

Assume we have 50% ammo save chance.

Each time a bullet is fired it has a 1/2 chance of being consumed and a 1/2 chance of being saved.

If we have a 30 bullet clip, then one would think that half the bullets would be saved, on average, resulting in an effective 45 bullet clip.

However this is wrong,

Imagine we have 100% ammo save chance. Then every bullet would have a 100% chance of being saved, meaning that a 30 bullet clip becomes 60 bullets right?

No, we both know this means infinite ammo. It's wrong because saved bullets can be saved again.

The chance of any bullet being saved twice is equal to the chance of it being saved once, and then once again. Since these events are independent, this probability is 1/4.

So instead what we want is how many times bullets can be saved before an amount of bullets equal to the clip size is not saved (i.e. we run out of ammo)

Luckily, there's a formula we can use.

The Negative Binomial Formula

Given,
p = the probability of an event X happening
r = the number of failures
k = the number of successes

The Negative Binomial Formula gives the probability of k successes happening before r failures.



And the expected value for this (i.e. the average number of successes) is



In order to calculate your own specific effective clip size, simply replace p for your ammo save chance as a decimal, (i.e. 47% = 0.47) and r for your clip size. Then add this number to your clip size, as it is the number of times you are saving bullets. Doesn't work if you insert 100%, as you'll end up dividing by zero. But in that case, you can figure it out yourself.

(formula images taken from https://en.wikipedia.org/wiki/Negative_binomial_distribution)