On time and on budget: why two 80% chances are not one
A 1,000-scenario worked example shows how to count joint deadline and budget success, and why multiplying two headline probabilities can mislead.
The promise usually contains two conditions
A fixed-price proposal often promises both a delivery date and a price. Suppose a hypothetical simulation says 80% of scenarios finish by day 30 and 80% cost at most $60,000. That does not mean 80% satisfy both conditions. You need to inspect the overlap.
Keep each scenario's duration paired with its cost. Count a scenario as a joint success only when its duration is at most 30 days and its cost is at most $60,000. Divide that count by the total number of simulated scenarios.
| Outcome | Count | Share |
|---|---|---|
| On time and within budget | 700 | 70% |
| On time, over budget | 100 | 10% |
| Late, within budget | 100 | 10% |
| Late and over budget | 100 | 10% |
The on-time total is 700 + 100 = 800. The within-budget total is also 800. But only 700 scenarios satisfy both. The chance of missing at least one condition is 30%, even though each individual miss rate is 20%.
Why multiplication is not a shortcut
Multiplying 0.80 by 0.80 gives 0.64, or 64%. That calculation is appropriate when the two success events are independent. In a project, longer duration can also increase labor cost, while one risk event can affect both schedule and spending. Those relationships can make independence an unsuitable assumption.
With two 80% marginal success rates, the joint success rate can lie anywhere from 60% to 80%. The lower bound follows because the two 20% failure groups could be completely separate; the upper bound occurs when both success groups coincide. The 70% example sits between those bounds. Neither the exact overlap nor its direction relative to 64% can be recovered from the two headline percentages alone.
Use the right financial condition
“Within budget” needs a precise definition. A cost ceiling of $60,000 differs from breaking even at a $75,000 sale price. It also differs from meeting a 25% margin target on that sale: the corresponding cost ceiling is $75,000 × 0.75 = $56,250.
If your commercial promise is a deadline plus a target margin, count scenarios whose duration meets the deadline and whose cost is at most $56,250. If the objective is simply avoiding a loss, use the price as the cost ceiling instead. Label the result with the condition you actually counted.
Preserve pairs when you inspect results
Do not independently sort the duration and cost arrays and then compare rows. That destroys the pairing between outcomes from the same scenario. Do not combine a deadline percentage from one project version with a cost percentage from another. A joint result requires both conditions to be evaluated against the same scenario set.
In a spreadsheet containing paired scenario data, a helper column can evaluate whether both limits are satisfied. For each row, return 1 for success and 0 otherwise, then average that column. A missing or invalid outcome needs investigation; silently dropping it can change the reported proportion.
Make a decision from the overlap
Compare alternative scopes, deadlines, and prices using the same definitions. A longer deadline might improve joint success without changing the financial ceiling. A higher price might improve a margin-based condition without changing physical delivery. A scope change can affect both and needs a fresh estimate.
These are modeled frequencies conditional on inputs and engine assumptions, not a guarantee about the next job. Read how dependencies change elapsed time and the model methodology. The deadline calculator answers the schedule-only question; it does not establish joint budget confidence.