Two Merging Paths and the 93% That Wasn't
A schedule has two paths converging on a single finish milestone. Both paths share the same initial activity, A (5 days, low variance). Path 1 continues through B and E: expected duration 40 days, σ = 4 days. Path 2 continues through C and F: expected duration 39 days, σ = 7 days. The internal target is day 46. The scheduler runs the probability calculation on Path 1 only, obtains z = (46 − 40) / 4 = 1.5, reads 93.3% from the normal table, and reports '93% on-time probability.' Leadership accepts this and removes the schedule risk reserve from the following quarter. Path 2 has not been analyzed at all, and its high-variance activity is a first-of-its-kind data conversion.
1. Evaluate the 93.3% figure. What is a defensible probability of hitting day 46, and why?
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The 93.3% is wrong - it is the probability of Path 1 finishing by day 46, not the project.
With converging paths, the project finishes when the last path finishes. If the paths were independent, P(project ≤ 46) = P(Path 1 ≤ 46) × P(Path 2 ≤ 46).
- Path 1: z = (46 − 40) / 4 = 1.50 → 93.3%
- Path 2: z = (46 − 39) / 7 = 1.00 → 84.1%
- Product: 0.933 × 0.841 ≈ 0.785, or about 78%
That 15-point drop is the whole point of the case: a near-critical path with large σ is not 'a path with float,' it is a second constraint. Path 2 has only 1 day less mean but nearly double the standard deviation, so its distribution overruns day 46 far more often. The scheduler's error was not arithmetic - it was scoping. Analyzing only the longest-mean path is a valid shortcut only when the other paths are dominated in both mean and variance.
Two caveats to state alongside the 78%:
- The product rule is optimistic here, because the paths are not independent - they share activity A. Any slip or quality problem in A affects both paths simultaneously, so the true probability is somewhat below 78%. Correlation makes convergence worse, not better.
- 78% is not a commitment. For an external commitment, day 46 is thin; the reserve was removed on the strength of a number that over-stated confidence by roughly 15 points.
2. Recompute for a tighter internal target of day 42 and explain what the result tells you about where to focus.
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- Path 1: z = (42 − 40) / 4 = 0.50 → 69.1%
- Path 2: z = (42 − 39) / 7 = 0.43 → 66.6%
- Combined (independence approximation): 0.691 × 0.666 ≈ 0.46, about 46%
What this shows: at day 46 the near-critical path shaved the probability from 93% to 78% - a real but survivable loss. At day 42 the two paths are roughly equidistant in standard-deviation terms, so both are limiting and the probability falls to under 50%. Tightening the target by 4 days costs about 32 probability points, and it does so almost entirely through the high-σ path.
Where to focus:
- Path 2's conversion activity is the single highest-leverage risk item - it has the largest σ on either path. A timeboxed pilot, a dry-run on a sample extract, or splitting the conversion behind a decision gate will cut σ more than any amount of staring at Path 1.
- Analyze every near-critical path whenever its σ is comparable to or larger than the margin between it and the critical path. The margin here is 1 day against a 3-day σ difference - clearly inside the noise.
- Do not buy confidence by only accelerating the critical path. Crashing Path 1 to a 37-day mean still leaves Path 2 at 39 with σ = 7, so the project's mean barely improves while the cost is real.
3. The data conversion turns out to be schedule-driven by the same vendor as another project. How does that change the interpretation of the 78%?
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It invalidates the independence assumption in the pessimistic direction - the 78% is now an upper bound, and probably a loose one.
Reasoning:
- The product rule assumes the paths fail independently. A shared vendor creates common-cause correlation: a vendor outage, staffing cut, or contract dispute delays the conversion and any other vendor-dependent work at the same time.
- Under correlation, the joint probability of both paths finishing on time is higher than the product of the marginals... but the project probability is driven by the maximum of the paths, and correlated tails make simultaneous overruns far more likely, so the project-level on-time probability falls below the independent estimate.
- Practically: at z ≈ 1.0 to 1.5 on both paths, independence gives roughly 78% for day 46; with strong correlation the effective figure behaves closer to the weaker path alone, i.e. nearer 84% for the shared-event case but with a much fatter joint tail - the honest statement is 'somewhere in the mid-70s at best, with a heavier downside.'
So what do you do differently:
- Do not multiply marginals and call it a project probability when paths share a resource, vendor, or predecessor. Model the shared driver as a single risk event applied to both paths.
- Manage the vendor as a project-level risk, not as two activity-level risks. One mitigation (contractual date, escalation path, alternate extract method) retires risk on both paths.
- Hold the reserve. The reserve was removed on the strength of a 93% that was already wrong; the correlation finding makes removal indefensible.
- Re-run the analysis with a correlation assumption (or a simple scenario: 'vendor slips 5 days' applied to both paths) and report the range, not a point.