Which Single Number Do You Report?

First, a quick retrieval check — a familiar trick, new setting.

A monitoring dashboard reports: “at least one of these two identical, independently-operating sensors is currently reporting bad data.” Each sensor independently has a 50% chance of reporting bad data on any check. What’s P(both are reporting bad data | at least one is)? (This is Lesson 5’s two-child puzzle wearing different clothes — the conditioning-on-how-you-learned-it trap, not plain independence.)

(Confirmed in the solution.)

You’ve built posteriors (Lesson 15), predicted from them (Lesson 18), and even hand-built one from a non-conjugate grid (Lesson 21). Often, though, you’re forced to report just one number — “how many bugs are left?”, not “here’s a whole distribution.” Three standard candidates:

• Mean: the probability-weighted average. Sensitive to a long tail — rare, severe outcomes pull it upward even if most outcomes are mild.
• Median: the value such that half the posterior’s probability lies at or below it, half above. Cares how much probability is in the tail, not how far the tail extends.
• Mode: the single most plausible value — wherever the distribution peaks. Ignores everything about the shape except the location of that one peak.

These are not three approximations of one “true” answer. Each is the number that minimizes a different penalty (loss function) for being wrong:

• Penalized by squared error, (guess − actual)² → the mean minimizes your expected penalty.
• Penalized by absolute error, |guess − actual| → the median minimizes it.
• Credited only for the exact right value (0/1 loss) → the mode minimizes it.

Your posterior, from a QA review, over the number of critical bugs remaining in a release:

(That last row is real — a small chance the release has a deeply nested issue that would surface as 10 distinct bug reports; QA teams see distributions shaped like this constantly: usually clean, rarely catastrophic.)

Part 1 — Mode. Which single value has the highest posterior probability?

Part 2 — Median. Add up probability from the bottom until you cross 0.5 — which value does the median land on?

Part 3 — Mean (the numeric answer above). Compute Σ (bugs × probability) across all five rows.

Part 4 — Which number do you actually report? You’re advising a release manager who has to decide whether to delay shipping. The cost of under-preparing for bugs scales badly — one 10-bug nightmare release costs far more than ten separate 1-bug releases, not just proportionally more. Given that, which of your three numbers (0, the median, or the mean) best reflects what the manager should actually weigh, and why is reporting the mode here actively misleading despite being the single most “likely” outcome?