We revisit the scaling properties of growing spheres randomly seeded in d =2 ,3 , and 4 dimensions using a mean-field approach. We model the insertion probability without assuming a priori a functional form for the radius distribution. The functional form of the insertion probability shows an unprecedented agreement with numerical simulations in d =2 ,3 , and 4 dimensions. We infer from the insertion probability the scaling behavior of the random Apollonian packing and its fractal dimensions. The validity of our model is assessed with sets of 256 simulations each containing 20 ×10<SUP>6</SUP> spheres in two, three, and four dimensions.