  In the brane world scenario, possessing the self-tuning property of the
cosmological constant, we study the probability amplitude for an initial state
of the Universe to transform to final states with some curvatures. In the Kim,
Kyae and Lee model, there exists a finite range of parameters such that the
transition amplitude to a near flat universe is exponentially dominated by the
smallness of curvature, > e^{positive number}/{Lambda-bar}^2}, i.e. is
dominated by an almost flat universe of |Lambda-bar|<epsilon.
