In this week's reading, Steven Pinker makes a claim about random processes: random events cluster. Streaks, clumps and coincidences are what randomness actually looks like. Before taking his word for it, let's do what this course is about: test the claim quantitatively.
One of these star fields was placed by a random number generator. The other was arranged to look random, the way most people imagine randomness. Click the one you think is genuinely random. Teammates in a shared session see the same skies and the same answer.
Humans expect randomness to be evenly spread. True randomness has no memory and no quota: nothing stops three stars landing in the same corner, so clumps and empty voids appear naturally. The 'too even' field is the imposter, each point was forced to keep its distance from the others.
This is Pinker's point. When we see a cluster (of shark attacks, cancer cases, basketball makes) we infer a cause. Often the cluster is exactly what chance predicts.
A fair coin is about to be flipped 100 times. First, commit to a prediction: how long will the longest run of identical outcomes be? Everyone in a shared session sees the same flips.
For n fair flips, the longest run is typically around log₂(n) + 1 ≈ 7.6 for n = 100. Most people guess 4 or 5. If you faked 100 flips by hand, your longest run would probably be too short, and a statistician could catch you.
Runs are one way to make the fuzzy word 'cluster' precise: a cluster in time is a run, and we can compute exactly how long runs should be under pure chance.
Each experiment below draws many samples of n coin flips and plots the share of heads in each sample. Drag the slider and watch what happens to the spread.
Small samples produce extreme results far more often. With n = 10, getting 80% heads is unremarkable; with n = 500 it would be astonishing. Clustering, and every other 'weird' pattern, is more pronounced in smaller samples.
This is why the counties with the highest cancer rates are mostly small rural counties, and so are the counties with the lowest rates. Small n, wild swings.
Clusters fool the eye; summary statistics can fool it too. A classic medical study compared two treatments for kidney stones. Here are the overall success rates:
| Treatment | Successes | Success rate |
|---|---|---|
| Treatment A (open surgery) | 273 of 350 | 78% |
| Treatment B (small puncture) | 289 of 350 | 83% |
| Group | Treatment A | Treatment B |
|---|---|---|
| Small stones | 93% (81/87) | 87% (234/270) |
| Large stones | 73% (192/263) | 69% (55/80) |
| All patients | 78% (273/350) | 83% (289/350) |
Treatment A wins in both subgroups, yet loses overall. Doctors assigned the hard cases (large stones) to A and the easy cases to B. Assignment was biased, so the overall comparison is not a fair fight. This is selection bias, and no amount of staring at the first table could reveal it.
This session previewed the toolbox we will build all year: logic, probability and statistics, algorithms and simulation, decision theory. Each tool exists to keep our judgments honest when intuition fails, as it just did, twice.