According to the PMBOK® Guide, specifically the Perform Quantitative Risk Analysis process, certain mathematical tools are used to evaluate uncertainty and make informed choices when faced with multiple paths.

Decision Tree Analysis: This is a diagramming and calculation technique used to evaluate several alternate courses of action. It uses Expected Monetary Value (EMV) to calculate the average outcome when the future includes uncertain scenarios.

Optimum Course of Action: By calculating the EMV for each " branch " of the tree (multiplying the probability of an event by its financial impact), the project manager can mathematically determine which path provides the highest value or the lowest cost to the organization.

Evaluation of Alternatives: It is particularly effective for " Make-vs-Buy " scenarios or " Upgrade-vs-Replace " decisions where different paths have different costs, risks, and potential rewards.

Why other options are incorrect:

Option A: Sensitivity analysis: This tool (often visualized as a Tornado Diagram) is used to determine which individual risks have the most potential impact on project outcomes. It identifies the " most sensitive " variables but does not help in choosing between different strategic paths.

Option B: Simulation: This usually refers to Monte Carlo analysis, which uses a computer model to simulate the project many times to show the probability of completing the project on a certain date or at a certain cost. It measures overall project risk rather than selecting between specific discrete alternatives.

Option D: Influence diagram: While these are used in risk analysis, they are graphical representations of situations showing causal influences, time ordering of events, and other relationships between variables. They help in modeling risk but are not the primary tool for calculating the " optimum course of action " among alternatives in the same way a Decision Tree is.