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The paper studies automatic verification of liveness properties with probability 1 over parameterized programs that include probabilistic transitions, and proposes two novel approaches to the problem. The first approach is based on a Planner that occasionally determines the outcome of a finite sequence of "random" choices, while the other random choices are performed non-deterministically. Using a Planner, a probabilistic protocol can be treated just like a nonprobabilistic one and verified as such. The second approach is based on γ-fairness, a notion of fairness that is sound and complete for verifying simple temporal properties (whose only temporal operators are ⋄ and □) over finite-state systems. The paper presents a symbolic model checker based on γ-fairness.We then show how the network invariant approach can be adapted to accommodate probabilistic protocols. The utility of the Planner approach is demonstrated on a probabilistic mutual exclusion protocol. The utility of the approach of γ-fairness with network invariants is demonstrated on Lehman and Rabin's Courteous Philosophers algorithm.