Apportionment is the problem of distributing ℎ indivisible seats across states in proportion to the states' populations. In the context of the US House of Representatives, this problem has a rich history and is a prime example of interactions between mathematical analysis and political practice. Grimmett [22] suggested to apportion seats in a randomized way such that each state receives exactly their proportional share of seats in expectation (ex ante proportionality) and receives either ⌊ ⌋ or ⌈ ⌉ many seats ex post (quota). However, there is a vast space of randomized apportionment methods satisfying these two axioms, and so we additionally consider prominent axioms from the apportionment literature. Our main result is a randomized method satisfying quota, ex ante proportionality and house monotonicity -a property that prevents paradoxes when the number of seats changes and which we require to hold ex post. This result is based on a generalization of dependent rounding on bipartite graphs, which we call cumulative rounding and which might be of independent interest, as we demonstrate via applications beyond apportionment. 3 We will revisit this result in Section 3 and show that, while Balinski and Young's theorem makes additional implicit assumptions, the incompatibility between quota and population monotonicity continues to hold without these assumptions. 4 A second shortcoming of deterministic apportionment methods is a lack of fairness over time: For example, if the states' populations remain static, a state with a standard quota of, say, 1.5 might receive a single seat in every single apportionment and therefore only receive 2/3 of its deserved representation. Using randomized apportionment, the long-term average of a state's number of seats is proportional to the state's average share of the total population. 5 In mechanism design, a similar approach extends strategyproofness to universal strategyproofness [28].