Given a simple directed graph D=(V,A), let the size of the largest induced directed acyclic graph (dag) be denoted by mas(D). Let D∈D(n,p) be a random instance, obtained by choosing each of the \binom{n}{2} possible undirected edges independently with probability 2p and then orienting each chosen edge independently in one of two possible directions with probabibility 1/2. We obtain improved bounds on the range of concentration, upper and lower bounds of mas(D). Our main result is that mas(D) \geq \lfloor 2\log_q np - X \rfloor where q = (1-p)^{-1}, X=W if p \geq n^{-1/3+\epsilon} (\epsilon > 0 is any constant), X=W/(\ln q) if p \geq n^{-1/2}(\ln n)^2, and W is a suitably large constant. where we have an O(\ln \ln np/\ln q) term instead of W. This improves the previously known lower bound with an O(\ln \ln np/\ln q) term instead of W. We also obtain a slight improvement on the upper bound, using an upper bound on the number of acyclic orientations of an undirected graph. We also analyze a polynomial-time heuristic to find a large induced dag and show that it produces a solution whose size is at least \log _q np + \Theta (\sqrt{\log_q np}).