What do the assumptions presented in the video look like with the additional (immediately less likely, I know) assumption that intelligent life is common?

Is it possible that it'd look something like this (the box being the universe and each ellipsoid being the expansion of probes from one civilization)? Basically, the probes from each civilization would expand out and eventually meet other civilization's probes and continually war to maintain territory.

Also, is it more likely that there isn't intelligent life given your assumptions or that we're just among the first intelligent species in the universe and we shouldn't expect to see anything surprising?