quite a bit, actually, however there is a such thing as subdivision companies of large studios... for example, the film "Little Miss Sunshine" was made by Fox Searchlight Pictures, which is but a small portion of 20th century Fox that specializes in "independent" movies. This is not an independent film, but a major studio film disguised as independent... Though, these types of films generally feature a lower budget than most movies, to give them an "indie feel" Little miss sunshine was made with a budget of $8 million, a DREAM for most real indie filmmakers.
So, calling that movie independent is an outright lie, which happens a lot, allowing these major studios to dominate the film festival scene, which was developed mainly to keep them away. Look at Fox Searchlight's filmography and you might see a lot of movies you formerly believed to be independent.
Though technically, in a way, they can be called independent, most people would agree that they aren't independent... just low budget.
So if we remove all the movies that are produced this way, then the answer to your question would be significantly lower.