🚌 What's This About
A bus runs on average every 10 minutes — sometimes after 4, sometimes after 18 minutes. The operator proudly
says: "mean gap 10 min, so mean wait time 5 min." But as a passenger, you arrive at the stop at a
random time, and it feels like you're always waiting longer. Who's right — the
operator or you?
Your prediction: The gap you land in is, on average, …
🔒 Guess first, then reveal — pick a prediction above to see the verdict, metrics, and frequency chart.
① The Schedule as a Timeline — Where Do Arrivals Land?
Vertical strokes = buses, bands in between = gaps (longer = wider). Triangles = your random arrivals; the most recently hit gap is highlighted.
—
② Try It Yourself — Arrive Randomly
Arrive at the stop at a random time repeatedly and watch how "your average" converges to the true expected value with every throw.
No arrival yet. Arrive at the stop at a random time — which gap do you land in?
Scheduled gap E[L]
(operator)
(operator)
—
mean of all gaps
Experienced gap
(passenger)
(passenger)
—
E[L²]/E[L]
Naive wait time
—
E[L]/2
Real wait time
—
E[L²]/(2·E[L])
③ Distribution of Gaps: Counted vs. Experienced
Blue = every gap counts once (operator's view). Orange = every gap weighted by its length (passenger's view) — the same gaps, but shifted right. Vertical lines = the two means.
Concepts
① The Symbols — What Stands for What?
The formulas below always use the same few quantities. L₁, L₂, … Lₙ are
the individual gaps between buses (n of them). T = L₁+L₂+…+Lₙ is the total schedule time.
E[L] is the plain ordinary mean of all gaps — what the operator calls the "scheduled gap."
E[L²] is the mean of the squared gaps — it shows up shortly because long gaps
weigh in disproportionately. Var(L) = E[L²] − E[L]² measures how much the
gaps vary. That's all we need — everything below builds only on these four quantities.
② Which Gap Do You Even Land In?
You arrive at a random point in time between 0 and T — not at a random
gap number. A gap of length L takes up the share L/T of the total schedule time, so:
P(you land in this gap) = L/T
A 12-minute gap is thus hit six times as often as a 2-minute gap — even though the operator counts
both exactly the same when computing their mean (each once).③ Where in the Gap Do You Land?
Suppose you've landed in a gap of length L — where exactly within it is
uniformly distributed. Sometimes you're right at the start (nearly the whole gap's wait), sometimes right
before the next bus (almost none). Averaged over many arrivals in equal-length gaps, the remaining wait time
is L/2 on average. If you land at minute 8 of 12, say, you only wait 4 minutes — that
doesn't contradict L/2 = 6, it's simply one single throw from exactly this spread.
④ Both Effects Combined — the Formula
The wait time weights every gap by its hit probability from ② and multiplies
by the mean remaining wait from ③:
E[wait time] = Σᵢ (Lᵢ/T) · (Lᵢ/2) = E[L²] / (2·E[L])
The "experienced gap" arises the same way — instead of weighting every gap by 1 like the operator does, you
weight it by its own length:
E[experienced gap] = Σᵢ Lᵢ·(Lᵢ/T) = E[L²]/E[L] = E[L] + Var(L)/E[L]
The last rearrangement shows directly: with no spread (Var = 0), the whole effect vanishes.⑤ More Spread ⇒ Bigger Effect
The surcharge in the formula above is exactly Var(L)/E[L]. With an exact
schedule (Var = 0), there's no paradox. With purely random (Poisson) arrivals, Var = E[L]² — then the experienced
gap is twice as long as E[L], and the wait time equals the entire mean gap
instead of half.
⑥ Worked Example
Four gaps: 2, 4, 6, 12 minutes (sum T = 24, operator's mean E[L] = 6 min).
The operator counts every gap equally ("share of n" column). But you hit them ∝ length ("hit prob." column) — the
12-min gap in half of all cases, even though it's only 1 of 4 gaps. Experienced gap:
E[L²]/E[L] = (4+16+36+144)/4 / 6 = 50/6 ≈ 8.3 min. Real wait time: E[L²]/(2·E[L]) = 50/12 ≈
4.2 min instead of a naive 3 min — not because you always wait until the very end in the 12-min gap
(there, too, the average is only 6 min), but because you land there so often in the first place.
| Gap L | Share of n | Hit prob. L/T | Mean rest L/2 |
|---|---|---|---|
| 2 min | 25 % | 8 % | 1 min |
| 4 min | 25 % | 17 % | 2 min |
| 6 min | 25 % | 25 % | 3 min |
| 12 min | 25 % | 50 % | 6 min |
Related Phenomena & Tools
The same math shows up wherever states are observed proportional to their duration or size,
instead of counted individually and equally: the class-size paradox (students experience larger
courses on average than the university reports — because more students sit in large courses), servers, restaurants,
or highways that seem constantly busy, and length-biased sampling in survival/screening studies
(cases "present" on a given cutoff date have overlong courses — correctable by weighting with 1/L if the
bias is known). Closely related: the friendship paradox — the same formula E[k²]/E[k], just for
network contacts instead of time intervals.
→ Friendship Paradox
→ Survivorship Bias