This paper proposes a tractable bi-level model to explore the mode choice of travelers and optimal operation decisions of a public transit operator and a ride-sourcing operator under a multi-modal transportation system. In such a multi-modal system, users may travel by bus, metro, and/or ride-sourcing service. The lower-level network model is utilized to capture the equilibrium mode choices with the consideration of travelers’ heterogeneity in terms of the value of time (VOT), where deterministic user equilibrium (DUE) and stochastic user equilibrium (SUE) are formulated and compared. In the upper-level model, subject to the DUE/SUE, the two operators optimize their operation decisions in the notion that the profit is maximized or the total system cost is minimized. The existence and uniqueness/non-uniqueness of the multi-modal mode choice equilibrium are analyzed. The optimal operation decisions are also compared under different operation regimes. Our results show that SUE model provides a more conservative estimate of the ride-sourcing operator's profit and total system travel cost compared to DUE. Besides, the total system cost can be reduced while the profit of the ride-sourcing company can be increased under appropriate operation decisions of the public transit operator and the ride-sourcing operator. Sensitivity analysis further indicates that travelers with longer total travel distance tend to use short-distance ride-sourcing services, which suggests that the ride-sourcing operator should dispatch more vehicles to suburbs located in remote areas to enhance service quality.