I've spent a long time searching but cant find any information. I'm probably searching the wrong terms.
Example: If 50% of chairs are red, and 25% of chairs are tall. What percentage of chairs are both red and tall.
I'd like to be able to do this for n variables.
Example: 3% of chairs have three legs. How many chairs are red, tall and three legged?
I know this would just be theoretical but let's assume a completely random distribution of traits.
In general it is impossible to say, but if they are completely randomly distributed and independent qualities as you say than it is just the product of the probabilities. For example, if $\frac12$ of chairs are red and $\frac14$ are tall, we expect $\frac14$ of the red chairs to be tall as well. So $\frac18$ of all the chairs should be red and tall. And $3\%$ of those should have $3$ legs, so $\frac3{800}$ of all the chairs should have all these properties.