In previous papers ([4] and [5]) E. Öztürk proved several results about the existence of rearrangements of series with certain properties. The purpose of this paper is to show that the rearrangements constructed in two of these theorems are rare, in the sense that the set of rearrangements having the desired property forms a set of the first Baire category in the set of all rearrangements. These results are related to an earlier theorem of this type by H. Miller [3].