A bus comes every 10 minutes on average, but with no timetable: the gaps are random, sometimes 2 minutes, sometimes 25. You walk up to the stop at an arbitrary moment. How long do you wait on average? Lock in your answer, then check it against a bus line, a simulation of 100,000 riders and the explanations.
A bus comes every 10 minutes on average, but with no timetable: the gaps are random, sometimes 2 minutes, sometimes 25. You walk up at an arbitrary moment and wait 10 minutes on average, a whole average gap. Long gaps take up more of the clock, so you are more likely to land in one. The gap between buses that you land in lasts 20 minutes on average.
In probability theory this puzzle is called the waiting time paradox or the inspection paradox. The setup is very simple.
In probability theory this puzzle is called the waiting time paradox or the inspection paradox: when you "inspect" the line at a random moment, you see not a typical gap but a longer one.
Buses on one line reach your stop every 10 minutes on average. Over five hours, from 7:00 to 12:00, 30 of them come by on average.
There is no timetable. Each bus arrives independently of the others, so the gaps vary a lot: sometimes two buses come 2 minutes apart, sometimes nothing shows up for 25 minutes. This kind of stream is called a Poisson process. A Geiger counter clicks with the same kind of randomness.
You don't check any app. You walk up to the stop at an arbitrary moment that has nothing to do with the buses, and you take the first bus that comes.
How many minutes do you wait for a bus on average?
"Half the gap, so 5 minutes"
In fact it is 10 minutes. Five minutes would be right for buses that keep a strict timetable. When the gaps are random, you land in long ones more often, and the wait doubles.
One hour of such a line might look like this: seven buses, gaps from 2 to 22 minutes.
Lock in your answer before reading on.
Lock in your answer before reading on. Then compare it with a timetable and with random buses.
A breakdown of your answer will appear here.
Each row is one hour, from 7:00 to 12:00. The small blocks are buses. Tap the line anywhere to "arrive" at the stop at that moment, or let chance pick the moment. The gap you landed in is highlighted in yellow, and the magenta arrow shows how long you waited.
The day's average gap is close to 10 minutes. But your "own" average gap, the average length of the gaps you landed in, is clearly longer. Switch to the timetable: every gap there is 10 minutes, and the wait drops to 5.
"Exact for this day" averages the wait over every moment from 7:00 to 12:00 for these particular buses. The more arrivals, the closer your average gets to it.
The computer builds a line of about 100,000 random buses, one every 10 minutes on average, and sends riders to the stop at random moments. For each one we record the wait and the length of the gap they landed in. Lines are the exact formulas, dots are the simulation.
Left: the wait has the same distribution as an ordinary gap between buses, with mean 10 minutes. Right: the gaps riders landed in (dots) are shifted to the right of all the line's gaps (grey line). Short gaps hardly ever show up, because they are hard to land in. Vertical ticks mark the means.
The slip is not in the arithmetic but in the sample. "The average gap" is computed over buses, while a rider sees a gap picked by time.
Four buses in half an hour give three gaps: 2, 3 and 25 minutes, 10 on average. Spread 12 riders evenly, one every 2.5 minutes. The short gaps get one rider each, the long gap gets ten. A rider picks a moment, not a bus, and the long gap takes up 25 of the 30 minutes.
So the gap you land in is chosen with weight proportional to its length. That is what length-biased sampling means.
| Gap | Chance to land | Wait inside |
|---|---|---|
| 2 | 2/30 | 1 |
| 3 | 3/30 | 1.5 |
| 25 | 25/30 | 12.5 |
Inside a gap you wait half of it on average. But the average has to be weighted by the chance of landing there: (2 · 1 + 3 · 1.5 + 25 · 12.5) / 30 = 319/30 ≈ 10.6 minutes. The gap you land in is (2² + 3² + 25²) / 30 ≈ 21.3 minutes long on average.
Half the average gap, 5 minutes, would only come out if all three gaps were 10 minutes.
For Poisson buses there is an even shorter argument. Buses are independent, so in every minute the chance of a bus is the same, however long ago the last one came. Already waited 10 minutes? The expected remaining wait is still 10.
So the moment you arrive changes nothing: from any point to the next bus is, on average, as long as from one bus to the next, 10 minutes. This property is called memorylessness. Among continuous distributions only the exponential has it.
Look both ways from the moment you arrive. Forward to the next bus: 10 minutes on average. Back to the previous one: also 10, because a Poisson stream played backwards looks the same. Together, the gap you are standing in lasts 20 minutes on average, twice an ordinary gap.
No contradiction: an ordinary gap really is 10 minutes on average, but you are not standing in an ordinary one.
A university runs five courses: four seminars of 10 students and one lecture of 160. The dean computes the average course size: (4 · 10 + 160) / 5 = 40. Now ask all 200 students how many people are in their course. 40 students say "10" and 160 say "160". The average answer is (4 · 10² + 160²) / 200 = 130.
It's the same arithmetic as with the buses: a big course shows up in the survey as many times as it has students, just as a long gap "catches" riders in proportion to its length. And in the friendship paradox a person with many friends appears on many friend lists, so your friends have more friends than you on average.
The wait equals half the average gap only when there is no spread (CV = 0). Each unit of CV² adds another half of the average gap. In a Poisson stream the standard deviation of a gap equals its mean, so CV = 1 and the wait doubles. When buses bunch up, the spread is even larger and you wait longer than the average gap.
Poisson buses are just one point on a scale. Real lines can be steadier, or worse. And biased sampling hides in many places besides bus stops.
The average gap is always 10 minutes. Only the spread changes. Gaps follow a gamma distribution with shape k: k = 1 gives Poisson buses, large k approaches a timetable, k < 1 gives buses that clump together. The curve shows 5 · (1 + CV²).
Even a line with a timetable rarely stays even. If a bus runs a little late, more people are waiting at each stop. Boarding takes longer, so it falls further behind. The bus behind it picks up fewer people, runs faster and catches up. In the end two buses arrive back to back, and then there is a long gap with none.
This is called bus bunching. For riders it is worse than plain random buses: with the same 10-minute average gap, the wait goes above 10 minutes. Slide to the right and watch the numbers.
The left end shows why transit agencies care about regularity. Going from random buses to a timetable cuts the average wait by 5 minutes without adding a single bus.
The top strip shows courses the way the dean sees them: each course gets the same space. The bottom strip shows them the way students do: a course's width is proportional to its number of students. Poke the top strip at random and you hit a typical course. Poke the bottom one and you most likely hit a big one.
The bottom strip is the same bus line: courses instead of gaps, students instead of minutes. The formula is the same too: the average a student sees is E[s²] / E[s], where s is the course size.
Both numbers are honest; they answer different questions. The dean asks how many students an average course has. The student asks how many people an average student shares a course with.
The mechanism is always the same: something long, big or crowded is more likely to end up in our sample, because we "inspect" the world at a random moment or through a random person.
| Situation | What we see | Why it's biased |
|---|---|---|
| Gym, flight, café | "It's always packed" | A busy hour holds more visitors, and each of them sees a crowd. Hardly anyone sees the empty room. |
| A prison survey | Long sentences | An inmate with a long sentence stays inside longer, so on survey day he is more likely to be there than someone serving a few months. |
| A hospital on a given day | Long stays | Same thing: a patient staying a month is present on 30 "inspection days", a one-day patient on just one. |
| Cancer screening | Slow tumours | A slow tumour lives without symptoms for a long time, so a routine check is more likely to catch it. This is called length time bias. |
| A race | Everyone is very fast or very slow | A runner at your own speed never passes you and you never pass them, so you barely notice them. The ones going by are those whose speed is very different. |
| Friends | Friends have more friends | A person with many friends appears on many friend lists. See the friendship paradox. |
The second volume of William Feller's classic textbook "An Introduction to Probability Theory and Its Applications" has a section called "Waiting time paradoxes". It works through buses with exponential gaps: a "symmetry" argument gives half the average gap, while memorylessness gives the whole gap. The second answer is the right one.
That is the title of Allen Downey's 2015 post on his blog "Probably Overthinking It". Using Purdue University's class-size data for 2013–14, he worked out that the average course has about 31 students, while the average student sits in a course of about 56.
In the same post Downey recorded the arrival times of 70 trains on Boston's Red Line between 4 and 5 pm over several days. The average gap between trains was 7.8 minutes, while the gap a rider lands in averaged 8.8 minutes. Trains run more evenly than Poisson buses, so the effect is smaller, but it is there.
The sociologist Scott Feld published "Why your friends have more friends than you do" in the American Journal of Sociology. The friendship paradox runs on the same mechanism as the buses: people with many friends get "inspected" more often because they are on many lists.
Bus bunching, when buses on one line arrive in groups, is a well-known transit problem. It is also called clumping, convoying or platooning. A late bus picks up more riders and falls further behind, while the one following catches up. So if you waited a long time and then two buses came together, it was most likely no coincidence.
In medical statistics, length-biased sampling goes by the name length time bias. Routine screening is more likely to find tumours that grow slowly and stay without symptoms for a long time. As a result, survival among screen-detected cases can look better even when screening itself does not extend lives.
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