This paper presents a combined method of Gaussian mixture model and random forest to compute membership probabilities of stars by using large, high-dimensional data sets. A significant advantage of this method is that it allows us to easily identify likely cluster members in large data sets starting from small training samples. As a benchmark, we select 40318 stars in the field of the open cluster NGC 6405 from the Gaia Data Release 2 (Gaia-DR2, Cat. I/345) by means of all five astrometric (positions, proper motions, and parallax) and photometric parameters. We use this combined method to determine likely cluster members in an eleven-dimensional parameter space. A total number of 518 high-probability (>=0.6) memberships are obtained, and the mean parallax and proper motion of the cluster are determined to be 2.171+/-0.005 mas (461+/-1 pc) and (<{mu}_{alpha}_cos{delta}>, <{mu}_{delta}_>)=(-1.357+/-0.023, -5.823+/-0.020) mas/yr, respectively. In addition, we quantitatively evaluate the relative importance of the parameters for membership determination and find that colors and magnitudes cannot be ignored in membership determination when using the RF method. Our results show that this combined method exhibits good performance in handling arbitrary high-dimensional and large data sets, such as Gaia-DR2, and it can also be used to investigate other open clusters.