If you have two different events with different (known) probabilities, what is the best way to compare the probabilities?
For example, the relationship between $0.5$ and $0.7$ is not the same as the relationship between $0.79$ and $0.99$. In both cases, the second number is $0.2$ more than the first; however, raising the probability of something from $0.5$ to $0.7$ will not have nearly as much of an effect as raising it from $0.79$ to $0.99$.
To me, it seems like there should be some sort of precise way to compare probabilities, but I have not been able to find one. I want to be able to quantify the comparison in a way that makes sense. Given two numbers, there are countless ways to compare them (subtraction, division, logarithms, etc), but I want to know which comparisons make the most sense in the context of probability.
My current idea is converting the probabilities to z-scores, so $(0.5, 0.7)$ becomes $(0,0.52)$, which is a gain of $0.52$. Also, $(0.79,0.99)$ becomes $(0.81,2.31)$, which is a gain of $1.5$.