Generally speaking: you can't predict this with some calculation. Just some of the reasons:
When you supply gases to a vacuum system it's like a conveyor belt, pushed by the pumping system. Each pump type and model has different efficiencies for different gases (and at different pressures).
When you sputter, the ionization efficiency of each gas is different (also depends on the pressure)
Let's assume that you sputter some metal such as Cr or Ti - the metallic atoms would be extremely likely to bind oxygen (if present, and also if not...), possibly bind to nitrogen (if present, in the absence of oxygen), and unlikely to bind to Ar - for example. This would dramatically change the partial process pressures from any prediction/calculation.
This is a tough question - but I'm not sure if it's the relevant question here. The gas ratio in the chamber is far from the composition of the film. I think the question should be a correlation between the relative flow rates/ratios at constant sputtering conditions - versus the composition of the resulting films. This can also be challenging to analyze (especially with light elements), but it's typically more relevant to your end application.
As Lior Kornblum said it is not an easy to get the partial pressures of Ar and O. By the way, you also has to know the pupming speed. But you can use RSD software package ( free download from http://draftugentbe.webhosting.be/index.php?p=137) to get the partial preassures of the chamber gages, as well as many other parameters. Also you can start with studying the Berg model of reactive sputter deposition