# designing an equation that compares two values and returns a probability

Given two values, I'm trying to come up with a formula that will return 50% if both values are equal, 25% if the first value is half the second, 75% if the second is half the first. In other words:

given (a=3,b=12) returns .125

given (a=3,b=6) returns .25

given (a=3,b=3) returns .5

given (a=6,b=3) returns .75

given (a=12,b=3) returns .875

(a and b will always be positive)

... the idea being that if a is half of b, it's 25%, if it's half of half, it's 12.5% or half of 25%, if it's half of half of half, etc - the numbers go down if a is greater, but up in the same way if b is greater.

I'd love to know what that equation would look like - I don't really know enough about how this stuff fits together to do much more than fiddle with it on my own. (not even sure what tags to apply to this)

• Are you sure the second example is right? I think you meant "given (a=3,b=3) returns .5" What do you want it to return beyond these cases, e.g., if one is a quarter or some other fraction of the other? There is clearly not enough information for a unique solution, but with some assumptions we can create an interpolation.
– Emre
Oct 19, 2011 at 22:47
• What (if anything) do you want the value of the function to be if $a\not=b$ and $\frac{a}{2}\not=b$?
– user5137
Oct 19, 2011 at 22:48
• you're right, @emre, misplaced a 6 there. added a couple other cases. Oct 19, 2011 at 22:56
• @jack - I don't know how to answer that - if a isn't equal to b, and half a isn't equal to b... it'd depend on what a is, but that doesn't answer your question. Oct 19, 2011 at 23:00
• @mattlohkamp - Well, you're the one trying to construct the function; you tell me. We want (using the new construction from the edit): $f(x,y)=0.5$ if $x=y$, $f(x,y)=0.25$ if $x=\frac{y}{2}$ and $f(x,y)=0.75$ if $2x=y$. Do you want $f$ to be defined for any other values of $x$ and $y$? If not, we're done. If so, what do you want to do? Incidentally, where are the values of 0.125 and 0.875 coming from?
– user5137
Oct 19, 2011 at 23:07

This function will reproduce your numbers (at least if $a$ and $b$ are positive):
$$f(a,b)= \frac{a}{2b} \text{ when } a \le b$$ $$f(a,b)= 1- f(b,a) \text{ when } a \gt b$$
You can write the latter case as $f(a,b)= \dfrac{2a-b}{2a}$ when $a \ge b$.
You can write both as $\dfrac{1}{2}+\dfrac{(a-b)\times \min(a,b)}{2ab}$.