Stochastic cell fate decisions occur frequently during animal development. In many cases, individual cells randomly choose one cell fate out of a limited repertoire of fates, but with the relative frequency of the different possible fates tightly controlled. It is not understood how signaling networks enable such cell-autonomous stochastic decisions and what network properties control the relative frequency of the resulting cell fates. To address these questions, we studied the differentiation of the C. elegans P3.p cell as a model system for a cellautonomous stochastic cell fate decision. We use a novel time-lapse microscopy technique to measure the single-cell dynamics of BAR-1/β-catenin and LIN-39/Hox, two key regulators of the network, while monitoring both the timing and outcome of the decision. Surprisingly, the timing of the decision is highly regulated, even in mutants that drastically alter the frequency of the cell fates. Using experimental data and modeling approaches, we find that a combination of variability in LIN-39 levels and variability in the timing of BAR-1/β-catenin signaling are noise sources that bias the stochastic decision. Our results highlight that a novel aspect of Wnt signaling can provide sufficient variation in a signaling network to enable a cell-autonomous stochastic cell fate decision in a multicellular organism.