Per-correct-answer cost of a two-tier router, misroutes included

The four cells. A router assigns each task to one of two tiers. Each task either belongs on the cheap tier or needs the premium tier; the share that belongs on cheap is the easy-task share, b. The router errs in two directions: up, sending an easy task to the premium tier (costly, never wrong under the default assumption), or down, sending a hard task to the cheap tier (cheap, wrong more often). Every task lands in exactly one of the four resulting cells, and the four traffic shares sum to one.

Dispatch shares. The share of traffic the cheap tier receives is the correctly routed easy tasks plus the mistakenly routed hard tasks. Its complement goes to premium. These two numbers tell you what the router does to your traffic without needing any price or pass rate.

Blended pass rate. Each cell contributes its traffic share multiplied by the tier's pass rate on that kind of work. The routed mix's pass rate is the weighted sum across all four cells. Compare it against the all-premium rate to see the correctness the routing gives up. The gap can reach zero or turn negative: our routing study measured this on short checkable tasks where the premium tier's refusal rate was higher, and the page carries that finding as its default assumption for the premium tier's pass rate on easy work. Put an interval on any of the four rates with the pass-rate CI calculator before entering them here.

Cost per correct answer. Divide the blended cost per attempt by the blended pass rate. The page computes this for three arms: the routed mix, an all-premium baseline (every task sent to the premium tier) and an all-cheap floor (every task sent to the cheap tier). The cost-per-successful-task calculator derives the same ratio for a single path through one tier; this page sums the paths a router produces.

The saving, stated twice. The per-attempt saving is the premium price minus the blended routed price per attempt. The per-correct-answer saving replaces those with cost per correct answer. The page shows both. The per-attempt figure divides by total tasks; the per-correct-answer figure divides by tasks that pass.

The quality gap. The all-premium blended pass rate minus the routed blended pass rate, converted to extra failures per 1,000 tasks so the quantity has a unit a builder can picture. Turn those failures into a dollar figure with the rework and downstream cost page, or test whether the gap sits inside your tolerance on the error-rate threshold page.

Break-even misroute-down rate. Both cost per attempt and pass rate are affine functions of the misroute-down rate, so cost per correct answer equals the all-premium baseline at one closed-form crossing. Which direction erases the saving depends on the inputs: the page states the direction beside the crossing. When the inputs reverse the relationship, the saving grows with the misroute-down rate and is gone below the crossing, not above it. The page shows the value when it falls inside zero to one and suppresses it with its reason otherwise. One figure to carry to a budget conversation.

The interval, and the boundary. When the audited-sample pair is filled, a 95% Wilson interval on the misroute-down rate drives two re-runs of the full model, one at each bound. The higher bound sends more traffic to the cheap tier, producing a cheaper cost and a larger quality gap: the two move together. This page prices an up-front split where every task runs once on the tier it was assigned. A cheap-first policy that falls back to the premium tier after a detected failure is a different instrument; the retry and fallback comparison already prices the recovery. Delays, backoff and jitter belong to the backoff and drain planner.

Formula: cost per correct answer = blended cost per attempt / blended pass rate