Most practitioners reach for a triangular or PERT distribution by default. They have a minimum, a most likely, and a maximum, fit a PERT or triangular, and move on. That is not always wrong. But there is a class of risk problem where the approach is structurally incorrect — and where the general discrete distribution, implemented in @RISK as RiskDiscrete, is the right tool.
This article explains why, then presents a realistic mining case study in which RiskDiscrete, combined with a lognormal impact distribution, produces outputs that a single continuous distribution can never generate.
What the General Discrete Distribution does
Rees, in his book Business Risk and Simulation Modelling in Practice (Wiley 2017), describes the general discrete distribution as a function that explicitly defines the probability attached to each outcome of a finite set of values. It is defined through a table: a set of values and a corresponding set of probabilities that must sum to one. In @RISK, the syntax is straightforward:
Values and probabilities defined as arrays. Probabilities must sum to 1.0.
Returns one value per iteration, sampled according to assigned probabilities.
The distribution is used when outcomes are not points on a continuum but structurally distinct states. Rees identifies three principal uses: modelling scenarios with only a few mutually exclusive outcomes (low, medium, high), approximating other distributions, and replicating discrete processes such as Bernoulli or binomial. That first use — scenario modelling — is where the function earns its place in serious risk models.
Rees also notes that the maximum discrete value can itself be a distribution — for example an unbounded lognormal — allowing the right-hand tail to be extended where needed. That matters for projects where catastrophic outcomes must remain in the model without being truncated.
Why two-layer models matter
The standard single-distribution approach collapses two separate questions into one. It asks: what is the range of possible outcomes? A two-layer model separates the question into what it actually is — first, which scenario are we in; and second, given that scenario, what does the impact look like?
That distinction matters in practice. Consider processing cost in a mining project. A triangular from $800/oz to $2,200/oz with a most likely of $1,100/oz produces a smooth, unimodal curve. But that curve is a fiction. The real model has three structurally different ore grade environments, each with its own cost economics. The aggregate distribution should be multimodal — and a triangular will never show that.
RiskDiscrete enforces the scenario layer. It forces the analyst to commit to probabilities for mutually exclusive states before modelling the impact within each state. That discipline alone improves the quality of the conversation with decision-makers.
It separates two genuinely different analytical tasks: assigning scenario probabilities — a judgement about what is likely to be true — and parameterising impact distributions, a technical exercise in calibrating ranges within a known regime.
Case Study: Ore grade regimes in an open-pit mining project
A feasibility study for a gold project must estimate all-in sustaining cost (AISC) per ounce across a 15-year mine life. The geological team has classified the resource into three distinct ore grade regimes that will determine processing economics.
| Regime | Condition | Probability | AISC Mean | AISC Std Dev |
|---|---|---|---|---|
| High Grade (HG) | Ore grade > 3 g/t | 0.20 | $950/oz | $80/oz |
| Base Case (BC) | Ore grade 1.5–3 g/t | 0.55 | $1,150/oz | $140/oz |
| Low Grade (LG) | Ore grade < 1.5 g/t | 0.25 | $1,600/oz | $280/oz |
These are not three points on a cost scale. They represent different extraction realities. Low Grade is not a bad version of Base Case — it is a different business, with different equipment, staffing ratios, and environmental management costs. Elevated reagent consumption, reduced recovery, and higher haulage compound simultaneously. No triangular distribution can capture that structural discontinuity.
Model structure: two layers
The model is built in two layers. The first is the regime selector.
Returns 1, 2 or 3 each iteration. Drives all conditional parameters below.
The second layer is a regime-conditional lognormal draw for AISC. The lognormal is appropriate because cost overruns within a regime are right-skewed: contractor inefficiency, reagent price spikes, and unplanned maintenance push costs upward with no symmetric downside. The active mean and standard deviation switch on the integer returned by the selector.
IF(B6=1,INPUTS!C13,IF(B6=2,INPUTS!C14,INPUTS!C15)),
RiskName("AISC_Draw"))
Parameters pulled from INPUTS. Active mean and std dev are conditional on the regime selector value.
What the output shows — and why it matters
The primary output is project NPV: gold price (a PERT distribution) minus the regime-conditional AISC draw, multiplied by annual production, discounted over the mine life at the project hurdle rate, less initial Capex.
The aggregate NPV distribution is right-skewed, with downside bounded and upside substantial. That is a useful result on its own. But the real value of the two-layer structure lies one step deeper.
Using @RISK's filter function to isolate NPV by regime value (B6=1, 2, or 3) produces three conditional distributions — one per regime — each with its own P10/P50/P90. That shows precisely how much of the downside tail belongs to Low Grade versus Base Case, and how High Grade shifts the entire distribution to the right. The aggregate histogram cannot show this; the filtered distributions do. For a client conversation about where to invest in geological confidence, that decomposition is the right tool.
The tornado chart confirms the point: the regime selector is the dominant driver of NPV variance, ahead of gold price, production, and discount rate. A model with a single triangular on AISC would have buried the geological regime risk inside a smeared cost range and pointed the tornado at the wrong variables entirely.
"The distribution is defined through a table, which shows the value and probabilities of each possible outcome. It would generally be used: to model scenarios and other situations with only a few outcomes (e.g. low, medium, high)."
Rees, M. Chapter 9, The Design of Risk Models — Principles, Processes and Methodology.How AI complements this type of model
The two tasks where AI genuinely accelerates the work are probability elicitation and parameter calibration.
On elicitation. Structuring a productive conversation with a chief geologist about regime probabilities is harder than it sounds. The analyst must surface implicit knowledge without anchoring the expert prematurely. Claude or GPT can draft the elicitation questions, structure the workshop agenda, and document the rationale in language defensible to an investment committee. The analyst retains full authority over the final numbers — AI structures the conversation, it does not run it.
On parameter calibration. Lognormal parameters for AISC by regime need grounding in analogous project data. AI can synthesise publicly available cost benchmarks from comparable operations, propose initial estimates, and flag where assumed values deviate from observed industry ranges. The research phase that once took days compresses into hours — but the analyst still decides.
What AI should not do is propose the regime probabilities directly. Those encode geological judgement and site-specific knowledge no language model can access. Allow a tool to set them without rigorous expert validation and the model's foundation is compromised, however correct the lognormal parameters are.
Process + Tools + AI = Augmented Results. Remove the process, and you have a fast way to produce a wrong answer.
M. Carmona — AI and Risk Analysis in Projects, Taylor & Francis.That division of labour is not a constraint — it is the correct design for a model that will actually support a capital allocation decision.
The takeaway
RiskDiscrete is not a niche function for edge cases. It is the correct tool whenever your risk problem is fundamentally about mutually exclusive scenarios before it is about ranges within a scenario. In mining, that means ore grade regimes. In energy, permitting outcomes or construction execution states. In infrastructure, demand scenarios or geotechnical classifications.
The two-layer architecture — discrete scenario selector plus continuous impact distribution — produces output that is more honest, more informative, and more useful to decision-makers than any single continuous distribution applied across the full problem space.
The accompanying Excel/@RISK model for the ore grade case study is available to download. It includes three sheets: INPUTS (all assumptions in blue, change here only), MODEL (simulation engine with full @RISK formula syntax), and INSTRUCTIONS (step-by-step run guide, settings, and a list of assumptions to replace before live use).
If you are building probabilistic models and not using Risk Discrete where scenario structure exists, you may be leaving information on the table.