Finding a function that transforms one probability into another

So consider the following scenario (this may look like statistics but the motivation is machine learning):

The probability of someone clicking on my ad when shown it on a web page is p(click).

The probability of a firefox user clicking on my ad when shown it on a web page is p(click|firefox).

The probability of a visitor in California clicking on the ad is p(click|california).

I need a way to estimate, given p(click), p(click|firefox), and p(click|california), the probability of a firefox user that is also in California clicking on the ad, ie. p(click|firefox,california).

My thinking is that if I could find some function F for which F(p(click))=p(click|firefox), then I could use this function to estimate p(click|firefox,california) by looking at p(click|firefox,california) ~= F(p(click|california)).

Of course, there are many possible forms for this function F, and a lot hinges on whether the user's choice of browser is independent of their geographic location (I suspect largely, but not completely).

My advantage is that I have a lot of data, so I can test a wide variety of variations on F to see which works best.

So a naive F would be: F(x) = x+C where C is a parameter we set such that F(p(click))=F(p(click|firefox)).

Now, this isn't very satisfactory because for some values of x, F(x) could easily end up being greater than 0 or less than 1 - unacceptable given that we're looking for a probability here.

Another F that still suffers from that problem would be F(x)=xC.

Ideally it would be great to have some function F which perhaps takes an additional parameter that represents the degree to which location and browser choice are independent of each-other. We could then search the parameter space to find what works best.

Can anyone offer any suggestions?

edit: improve notation