The presence of copies in the museum exhibition is due to the fact that not all items can be exhibited for a long time (due to the material (paper, fabric), state of preservation, conditions of exposure, etc.). Usually, at the same time, it is indicated in the label that you have a copy in front of you, not an original.
It is difficult to judge the percentage of copies to originals, because this may depend on many factors: the museum's collection, the preservation of certain items, the profile of the museum ... But often in the museum's exposition, most of them are the originals.
But the Pushkin Museum. Pushkin, for example, contains in its exposition a large number of casts and copies, since IV Tsvetaev originally conceived the museum as a collection of casts and copies. But over time, even such items become cultural heritage and form a special cultural environment.
The more competent a team of art critics works in a museum, the more chances that the museum's collection will contain more genuine works of art, no matter if it is a modern museum or a historical one. The collection of the NCCA, for example, is collected mostly from the gifts of the artists themselves, who have known the artistic director of the NCCA for a long time and from the very best side, Leonid Bazhanov. Of course, it is just as important that the larger and more reputable the museum, the more funding it has and the more chances it will buy the desired originals. Which is always written on the label: "original". Or, which also happens - "copy". But the label under the painting always indicates this.