Randomized Hough transform (RHT) has been recently proposed as a new and efficient variation of the Hough transform for curve detection. In this paper the RHT is combined with the fuzzy cell Hough transform (FCHT) and a new variation, the randomized fuzzy cell Hough transform (RFCHT) is proposed. The p-dimensional parameter space of Hough transform is split into fuzzy cells with overlapped intervals of confidence. The fuzzy cells are defined as fuzzy numbers. The RFCHT selects p pixels from an edge image by random sampling and solves the p parameters of a curve. Then the p parameters accumulate to more than one fuzzy cells, since the fuzzy cells intervals are overlapped by adding a value that belongs in the interval [0, 1] and is calculated from the membership function of the corresponding fuzzy cell. The procedure continues by using a percentage of the total contour pixels. The RFCHT algorithm preserves the advantages of RHT and FCHT. The algorithm has good computational speed and small storage requirements due to random sampling and correct and more accurate detections, especially in noisy images, due to fuzzy cells.