If you are wanting to generate a second or third color by overlapping shapes as in your example, and want to do so accurately as it would happen in an RGB color system. You can use the transparency modes in illustrator. 'Overlay' and 'screen' will be your best options for replicating light-based color behavior.
see image below.
(tip: you might need to use a black background for the shapes in "screen" transparency mode)
If you want a mixture as far as paint would behave (not light-based colors.) then you can use 'multiply' transparency mode.
If you want to use the primary colors you should probably adjust them to cyan blue, magenta and yellow. But I kept your colors anyway.
Finally, if you wish to color pick the overlapping areas you probably will be disappointed when you realize the color picker will just copy the properties of one of the shapes (whichever is over the other). So my advice is to rasterize the image and then color pick the overlapping areas.
To rasterize the image:
- select all three circles with the transparency mode applied
- Go the menu 'Object' and choose 'Rasterize'
( make sure you duplicate your circles before you do this if you want to keep the original vector)