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State the fictional two-stage model
A project either receives a required approval with probability 45% or does not with probability 55%. Given approval, revenue outcomes are 30, 70 and 120 million with probabilities 20%, 50% and 30%. Without approval, revenue is zero with probability 80% or 30 million with probability 20%. These are stipulated learning inputs, not a valuation of an actual project.
Calculate each branch’s expected revenue
Given approval, the expectation is 0.20×30 + 0.50×70 + 0.30×120 = 77 million. Without approval, it is 0.80×0 + 0.20×30 = 6 million. Each calculation uses probabilities conditional on its own branch.
Apply the outer scenario weights
| Branch | Branch probability | Conditional mean | Contribution |
|---|---|---|---|
| Approval | 45% | 77 million | 34.65 million |
| No approval | 55% | 6 million | 3.30 million |
| Total | 100% | 37.95 million |
Check one terminal path
The approval-and-120-million path has probability 45%×30% = 13.5%. Its contribution to unconditional expected revenue is 13.5%×120 million = 16.2 million. Repeating that calculation for all five terminal paths produces the same 37.95 million expectation. Neither simply averaging 77 and 6 nor omitting the no-approval branch answers the stated model.