Let denote the power set of [n], ordered by inclusion, and let denote the random poset obtained from by retaining each element from independently at random with probability p and discarding it otherwise. Given any fixed poset F we determine the threshold for the property that contains F as an induced subposet. We also asymptotically determine the number of copies of a fixed poset F in . Finally, we obtain a number of results on the Ramsey properties of the random poset .