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Where the Chain Breaks

An error rate per step looks small. Across a chain of steps it does not stay small. This page runs the arithmetic for the simple case and for the real one, where steps differ and checks sit in between, and gives you a link to the result.

Part 1 · The simple case

Every step equally good, nothing checked

The calculation you can do in your head, and the reason long autonomous chains become uncomfortable. Set the accuracy of a single step, the length of the chain, and how many failed runs the business owner would accept.

Accuracy per step98.0%

An error rate of 2.0 percent. The slider is logarithmic: the nines are on the right.

Steps in the chain10

One to 100 steps in sequence.

Failed runs the business accepts

The standard the process owner would measure the chain against.

10 steps at 98.0 percent each: 18.2 percent of runs fail somewhere in the chain.

Chain completes

81.8%

every step correct

Chain fails

18.2%

at least one step wrong

Longest chain within tolerance

2steps

before more than 5 percent of runs fail

your error rate, 2.0 percenthalf the error, 1.00 percentdouble the error, 4.0 percenttolerance

The address bar always carries your settings.

Part 2 · The real case

Mixed steps and checks in between

In practice every step has its own accuracy. Some are deterministic, a rule engine or a line of code, and carry an error rate of zero. At a few points a check looks at the case so far. A check does not remove error for free: it catches a share of what is there and costs an intervention for each flag. The table runs all of it. The example is a client instruction on its way to a booking. Overwrite it with your own chain.

probabilistic step, sized by error deterministic step automated check human review error carried, with checks the same chain without checks

An error does not fade on the way. It rides along, step after step, until a check sees it. The curve shows the ride, and every step after the error works on a case that is already wrong.

#	Error rate	Catch rate
1	%	%
2	%	%
3	%	%
4	%	%
5	%	%
6	%	%
7	%	%

Error rate zero means a deterministic step. A check that sees this step only, a schema or a rule, catches nothing from earlier steps. A check that sees the whole case, a person or a reconciliation against the source, does.

Where 1'000 cases end up

86.6%

11.7%

866 run clean. No step made an error.

117 are caught. At least one error, every one of them stopped at a check.

17 leave wrong. An error that no check saw. The number the business and the auditors mean.

Undetected errors

17per 1'000 cases

1.7 percent of runs leave the chain wrong

Human touches

1'029per 1'000 cases

flags from automated checks, plus every case at a human review

Without any checks

134per 1'000 cases

the same steps, nothing in between

LinkedIn takes a link and nothing else, so it shows the page card with your chain behind it. To say something with it, copy the post text and paste it above the link.

What moves the number

Keep the chain short

Length works exponentially, accuracy per step only linearly. Merging two probabilistic steps into one does more than improving a model by a point.

Stay deterministic where you can

A rule or a line of code brings an error rate of zero into the chain and shortens its effective length. Set a step to zero in the table and watch. This is the 1.0 multiplier.

Put checks where errors are expensive

A check lowers the residual of every step before it and costs touches. Early in the chain it protects more steps, late in the chain it sees more of the damage. An automated check with a person on the flags scales. A person on every case does not. And a flag is never free: a chain whose flags all become manual cases has rebuilt the manual process for its hardest cases.

The arithmetic and its assumptions

Part 1. With an error rate p per step and n steps, the chain completes with probability (1 − p)ⁿ and fails with probability 1 − (1 − p)ⁿ. The longest chain within a tolerance t is the whole number part of ln(1 − t) / ln(1 − p).

Part 2. An error made at step i survives when it passes every later check, with probability equal to the product of (1 − c_j) over all checks j at or after i. A case runs clean with probability equal to the product of (1 − p_i). It leaves without an undetected error with probability equal to the product of (1 − p_i · survive_i). Caught is the difference between the two. Undetected is one minus the second product.

Touches. An automated check hands every flag to a person, so its touches are the probability that it fires. A human review looks at every case, so its touches are one per case. A check flags an error from an earlier step only if that error passed the checks in between.

Scope. A check that sees this step only catches errors made in that step. A check that sees the whole case catches errors from every earlier step that no check has caught yet, at the same rate as its own step's errors. That is generous for a person late in the chain, where an upstream error has long since turned into a plausible looking case.

Assumptions. Steps are independent, a caught error is treated as corrected, and checks add no errors of their own. The work that correction takes, a fix in place, a rerun from the failing step, or a manual takeover of the case, is outside this calculation. The figures are inputs, not measurements. The calculation becomes evidence only with error rates measured in your own pilot.