Electric vehicle charging facilities offer electric charge and parking to users for a fee. Both parking availability and electriccharge capacity are constrained resources, and as the demand for charging facilities grows with increasing electric vehicleadoption, so too does the potential for exceeding these resource limitations. In this paper, we study how prices set by thecharging facility impact the likelihood that resource constraints are exceeded. Specifically, we present probabilistic boundson the number of charging spots and the total power supply needed at a facility based on the characteristics of the arrivingvehicles. We assume the charging facility either offers a set of distinct and fixed charging rates to each user or allows the userto decide a charging deadline, from which a charging rate is determined. Users arrive randomly, requiring a random amountof charge. Additionally, each user has a random impatience factor that quantifies their value of time, and a random desiredtime to stay at a particular location. Assuming rational user behavior, and with knowledge of the probability distributionof the random parameters, we present high-confidence bounds on the total number of vehicles parked at the station and theaggregate power use of all vehicles actively charging. We demonstrate how these bounds can be used by a charging facility todetermine appropriate prices and investigate through a Monte-Carlo simulation case study the tightness of the bounds.