LEARN. PRACTICE. GROW.Start with what the number means
Before calculating, identify the event, observed population, period, and unit. Ten failures in one hundred executions of the same batch describe an observed frequency in that set; they do not establish failure probability for any future application. An expert may estimate a 20% probability while having little confidence in the evidence. Uncertainty about the estimate is not resolved by replacing 20% with an undefined confidence number. Ask for conditions, counterexamples, and data that could change the estimate. In a distribution of alternative outcomes, check that branches cover all considered possibilities without overlap. Probabilities of events that can coexist are not, by themselves, probabilities of exclusive branches. Record what remains unknown and which decision depends on that gap.
Rank without turning labels into money
An ordered scale helps distinguish levels using agreed criteria. If values 1, 2, and 3 are only categories, 3 does not mean three times the loss of 1. A matrix can establish priority without producing an expected cost in euros. Retain underlying physical or financial impact where available, and do not add ratings as if they were money. Also consider when the response must begin. A risk that may occur in six weeks can require a decision today because its alternative takes six weeks to prepare. Another risk may occur tomorrow with a response already available. Final ordering needs to combine explicit criteria and deadlines, rather than simply sorting by event date or report color.
Move from individual risks to joint outcomes
Two tasks may each meet their deadline with 80% probability without offering an 80% chance of joint success. If independent and both required, the result is 0.8 × 0.8 = 64%. If they depend on the same supplier window, independence needs review. Do not invent missing dependence: state the assumption and seek evidence. Adding expected losses remains valid for additive losses even with dependence, but the total distribution and its extremes may change. Avoid counting the same business loss twice when several components fail in one event. The sponsor needs to understand exposure of the complete objective; a list of individual ratings does not automatically answer that question.
Check whether the decision changes
Use sensitivity analysis to find assumptions that can change the choice. In an exercise measured in thousands of euros, A costs 4 plus 10% of loss L; B costs 30% of L with no fixed cost. Equating 4 + 0.1L with 0.3L gives a break-even loss of L = 20. Below that value B has lower expected cost; above it A does. If credible loss ranges from 15 to 30, an expected-cost choice is not robust across the whole interval. Improve evidence or state the recommendation’s condition. Check mandatory criteria before comparison: a cheap option breaching a constraint does not become admissible through averaging. These numbers are fictional, not financial advice or a PMI rule.
Enumerate a small schedule
In the fictional model, A and B start in parallel and final validation takes ten minutes after both finish. A takes twenty minutes with probability 0.7 or sixty with 0.3. B takes forty with 0.8 or eighty with 0.2. Under independence, the four pairs have probabilities 0.56, 0.14, 0.24, and 0.06. Calculate each duration as the maximum of A and B plus ten: outcomes are fifty, ninety, seventy, and ninety minutes. Mean duration is 62.8; P80 is seventy and P90 ninety, using the smallest outcome reaching the requested cumulative probability. Enumerating every branch of this finite model avoids sampling error. It does not remove uncertainty about chosen probabilities or turn the exercise into an empirical forecast.
Keep marginals and change dependence
Compare a second assumption: A/B pairs are 20/40 with probability 0.7, 60/40 with 0.1, and 60/80 with 0.2. Individual A and B probabilities remain unchanged, but slow states coincide more often. Mean maximum-plus-ten duration becomes sixty minutes; P80 remains seventy and P90 ninety. The probability that both are slow rises from 0.06 to 0.2. Thus a dependence change can move metrics in different directions: do not summarize everything as “more correlation increases every result.” In the first model, A alone determines parallel completion in 24% of branches; in the second, in 10%. These values help interpret model paths but do not demonstrate a real causal relationship or the best action to fund.
// Original finite educational model: exact enumeration, not Monte Carlo or production forecasting.
function summarize(rows) {
if (!rows.length || rows.some(r =>![r.a,r.b,r.p].every(Number.isFinite) || r.a<0 || r.b<0 || r.p<=0 || r.p>1) || Math.abs(rows.reduce((s,r)=>s+r.p,0)-1)>1e-12) throw Error('Invalid joint distribution');
const outcomes=rows.map(r=>({...r,minutes:Math.max(r.a,r.b)+10})).sort((a,b)=>a.minutes-b.minutes);
function quantile(p){if(!Number.isFinite(p)||p<=0||p>1)throw Error('Invalid quantile');let total=0;return outcomes.find(r=>(total+=r.p)+1e-12>=p).minutes;}
const round=x=>Math.round(x*1e10)/1e10;
return {mean:round(outcomes.reduce((s,r)=>s+r.p*r.minutes,0)),p80:quantile(.8),p90:quantile(.9),by70:round(outcomes.filter(r=>r.minutes<=70).reduce((s,r)=>s+r.p,0)),aControls:round(rows.filter(r=>r.a>r.b).reduce((s,r)=>s+r.p,0)),bothSlow:round(rows.filter(r=>r.a===60&&r.b===80).reduce((s,r)=>s+r.p,0))};
}
const independent=[{a:20,b:40,p:.56},{a:20,b:80,p:.14},{a:60,b:40,p:.24},{a:60,b:80,p:.06}];
const dependent=[{a:20,b:40,p:.7},{a:60,b:40,p:.1},{a:60,b:80,p:.2}];
console.log(JSON.stringify({independent:summarize(independent),dependent:summarize(dependent)}))In a fictional batch, migration and reconciliation each have an 80% chance of success. Under independence, meeting both has 64% probability, below the educational criterion of 75%. Changing that conclusion requires revisiting data or dependence, not averaging the two 80% values.
Common pitfalls
Confusing confidence with probability, adding categories as euros, multiplying without independence, duplicating losses, and recommending an option without testing plausible assumption ranges.
Related topics: Triggers, data, and common dependencies · Quantitative decisions and value of information · Effectiveness evidence and APS handover
Useful analysis connects data meaning to the decision. State units, dependencies, criteria, deadlines, and conditions under which the recommendation changes.
Reference: Integrating risk analysis and prioritization: a practitioner’s tool · PMI-RMP five-domain ECO, updated-2024 public document