Think twice / 10

The Red Stamp

The detector is right nine times out of ten. A crate gets a red stamp. How worried should you be?

How to play

You are checking a museum’s incoming shipment. Read how many crates are fake and how the detector treats each kind, then press Open the red-stamped pile.
Look at the two counts inside that pile. You can also Check the unstamped pile to see what escaped the warning. Each square represents one actual crate in this invented shipment.
Choose More fakes or Half and half and open the new shipment. The detector keeps exactly the same hit and false-alarm rates. Compare what a red stamp means in each setting.

What to notice

A museum buys a thousand small objects. A scanner flags the ones it thinks are forgeries. You are told it correctly flags nine out of every ten fakes and correctly leaves nine out of every ten genuine objects alone. A red stamp arrives on your desk. It feels like almost a verdict.

But there are two ways into that red pile: be one of the fakes the scanner catches, or be one of the genuine objects it wrongly flags. If there are many more genuine objects to begin with, a small fraction of that large group can outnumber the caught fakes. The scanner has not become worse. The pile you are asking about is a mixture of different starting groups.

This is the base-rate problem: how common something was before the signal helps determine what the signal means. Researchers Gerd Gigerenzer and Ulrich Hoffrage studied how presenting such problems as counts can help people reason through them. Here you can open the pile and literally count its two ingredients instead of treating “90% right” as the answer to every question.

Change only the shipment. When fakes are rare, a warning can mostly point to genuine objects. When fakes are common, the same warning carries a different weight. Also look at the unstamped crates: reassuring evidence is not certainty either. Real checks have costs, context and imperfectly known rates; this room fixes those rates so you can inspect one relationship clearly. Before trusting a warning, which starting population should you be asking about?

A little further: Gigerenzer and Hoffrage’s 1995 paper on frequency formats, and the Max Planck research report discussing natural frequencies. The museum and its shipments are invented for this room.

The museum’s receiving desk

Same detector, every shipment

Choose what arrives. The detector’s rules stay fixed.

Incoming: 10 fake and 990 genuine crates.

For every 10 fakes:
9 get a red stamp. 1 slips through.

For every 10 genuine crates:
9 pass unmarked. 1 gets a false alarm.

108

crates have a red stamp.

How many are actually fake?

Unpacking the local desk.

Shipments you have opened

One comparison per shipment. Reopening one does not add another thousand crates.

Same detector: catches 90% of fakes; falsely flags 10% of genuine crates. These are exact teaching counts, not measured accuracy.
Shipment	Fakes in all 1,000	Fake / genuine in stamped pile	Stamped crates that are fake
Open a stamped pile to start comparing.

Rules & model

Three fixed shipments

Each shipment contains exactly 1,000 crates. Rare fakes has 10 fake and 990 genuine; More fakes has 100 fake and 900 genuine; Half and half has 500 fake and 500 genuine. These facts are known to you as the observer. The detector does not change its behaviour when you select another shipment.

What the detector does

Exactly nine out of each ten fake crates get a red stamp. The remaining fake crate is unmarked. Exactly one out of each ten genuine crates gets a red stamp by mistake. The remaining nine are unmarked. Every crate belongs to exactly one of these four groups. All shipment sizes are multiples of ten, so none of these counts needs rounding. There is no randomness, second test, generated evidence or secret difficulty adjustment.

Opening a pile

The stamped pile contains caught fakes plus falsely flagged genuine crates. Its fake fraction is caught fakes divided by that pile’s total, not divided by all fakes or all crates. Rare fakes produces 9 fake plus 99 genuine, so 9 of 108 stamped crates are fake: 8⅓%, displayed as 8.3%. More fakes produces 90 fake plus 90 genuine, so 50% are fake. Half and half produces 450 fake plus 50 genuine, so 90% are fake. In every shipment, 900 of all 1,000 classifications are correct.

The unstamped pile contains missed fakes plus correctly cleared genuine crates. For Rare fakes, that is 1 fake plus 891 genuine. One square is one crate; the grid groups kinds together to make counting easier. It is not a random arrangement or an ordered arrival log. Percentages are rounded to one decimal place only for display.

What a warning cannot settle

This example assumes the starting proportions and both conditional detection rates are exactly known. Real settings may not supply those facts or keep the rates stable. Choosing whether to investigate also depends on costs and consequences, which this model does not assign. This is not medical, fraud-screening or security-system advice. It is an invented counting experiment, not a measurement of your reasoning ability. Up to three shipment comparisons stay in page memory for this visit. Nothing is uploaded or stored after you leave; all calculation happens on your clicks.