🧮Conjunction Fallacy Test

Can a detailed story make a logically narrower possibility feel more likely than the broader possibility that contains it?

Take this free Conjunction Fallacy Test, inspired by the famous Linda problem. You will read six short descriptions of fictional people and choose which of two statements seems more probable.

One option states a single condition.

The other combines that condition with an additional condition.

Under standard probability logic, a conjunction cannot be more probable than one of the events that contains it.

How to Take the Conjunction Fallacy Test

1. 1Press Start Test.
2. 2Read each person's description.
3. 3Compare the two answer choices.
4. 4Choose the statement you believe is more probable.
5. 5Complete all 6 scenarios.
6. 6Review how often you selected the combined statement.

Try to answer the probability question rather than:

“Which option sounds more like this person?”

That distinction is the heart of the experiment.

What Is the Conjunction Fallacy?

The conjunction fallacy occurs when people judge:

A and B

as more probable than:

A

alone.

Under standard probability theory:

P(A and B) ≤ P(A)

The conjunction can equal the probability of A in a special case where B always occurs whenever A occurs.

But it cannot exceed A.

Why?

Because every case in which A and B are both true is already included among the cases in which A is true.

Adding another requirement cannot create more possible cases.

The Famous Linda Problem

Psychologists Amos Tversky and Daniel Kahneman published the landmark conjunction-fallacy paper in 1983.

Participants read a description of a fictional woman named Linda.

The description made her seem strongly associated with social-justice and feminist concerns.

Participants then compared statements including:

Linda is a bank teller

and

Linda is a bank teller and active in the feminist movement

Many judged the combined statement as more probable.

Yet every feminist bank teller is also a bank teller.

The combined category is therefore a subset of the broader category.

The finding became one of the most famous examples of conflict between intuitive plausibility and formal probability.

Why the Combined Option Feels Better

The detailed option fits the story more closely.

That is where representativeness becomes important.

People often judge probability partly by asking:

“How well does this outcome match the description?”

The feminist-bank-teller option matches Linda's profile better than the plain bank-teller option.

But similarity to a story is not the same as probability.

A highly representative outcome can still be mathematically less probable because it contains extra conditions.

The fallacy occurs when that intuitive fit overrides the conjunction rule.

The Set Logic Behind the Rule

Imagine 1,000 people.

Some are bank tellers.

A smaller subset are both:

bank tellers AND feminist activists

Every member of the smaller group must also belong to the larger bank-teller group.

Therefore:

number of bank tellers and activists ≤ number of bank tellers

The same structure applies to any conjunction.

For example:

teachers who run marathons

cannot be more numerous than:

teachers

because the first group is contained inside the second.

This set relationship is the simplest way to see why the conjunction rule works.

But Natural Language Creates an Important Complication

The mathematics is strict.

Human language is not always.

Researchers have debated the Linda problem partly because words such as “probable” and “and” can be interpreted conversationally in more than one way.

Some participants may read:

“Linda is a bank teller”

as implicitly meaning:

“Linda is a bank teller and not especially connected with feminism.”

That is not the formal logical meaning intended by the experiment.

Research by Hertwig and colleagues has shown that wording and interpretation can influence conjunction responses.

This does not eliminate the conjunction fallacy as a research phenomenon.

It does mean that the cleanest interpretation separates:

the formal probability rule

from

the pragmatic way ordinary language is understood.

Frequency Formats Can Reduce the Error

Another important finding is that people often reason better when the problem is presented using natural frequencies.

Instead of asking:

“Which is more probable?”

you might ask:

“Out of 100 people matching this description, how many would be bank tellers, and how many would be bank tellers who are also feminist activists?”

The subset relationship becomes easier to visualize.

Research has found that frequency-based representations can markedly reduce conjunction errors, although they do not remove them in every procedure.

This suggests the format of probability information matters.

Does Statistical Training Eliminate the Fallacy?

No—but the strong claim that training never helps is also too simple.

Tversky and Kahneman found conjunction errors even among statistically knowledgeable participants in some conditions.

Later research shows that performance depends on:

• ✓wording;
• ✓response format;
• ✓explicit probability instruction;
• ✓frequency representation;
• ✓statistical knowledge;
• ✓and how transparent the subset relationship is.

Formal knowledge can help when people actually apply it.

The effect is therefore not proof that humans are incapable of probability reasoning.

It shows how intuitive reasoning can compete with formal rules.

What Does Your Score Mean?

Your site gives you six scenarios.

If you choose the conjunction on four of them, that means:

4 of your 6 responses violated the conjunction rule under the intended formal interpretation.

It does not mean:

“You are 67% irrational.”

Six items are too few to estimate a stable reasoning trait.

The scenarios also vary in how convincing their descriptions are.

Use the score as a compact demonstration of intuitive probability judgment.

Conjunction Fallacy vs. Base-Rate Neglect

These effects are related through the broader concept of representativeness but are not identical.

Conjunction fallacy: a combined event is judged more probable than one of its components.

Base-rate neglect: relevant prior probabilities are underweighted when vivid case information is available.

The Linda description can encourage both kinds of reasoning, but the conjunction rule has a particularly clear mathematical constraint.

Conjunction Fallacy vs. Gambler's Fallacy

The Gambler's Fallacy Test concerns mistaken beliefs about sequences of independent random events.

The conjunction fallacy concerns probability relationships between overlapping categories or events.

One asks:

“What happens next after a streak?”

The other asks:

“Can A-and-B be more likely than A?”

These are different errors.

Why Stories Are So Powerful

A coherent narrative can feel more convincing than a bland statistical statement.

That is useful in ordinary life because stories help us organize information and infer likely causes.

But a compelling story can become dangerous when it quietly replaces probability structure.

A detailed prediction may sound realistic because every detail fits.

Yet every added condition usually makes the full prediction less likely.

For forecasting, that creates a useful discipline:

separate narrative coherence from mathematical probability.

How to Avoid the Conjunction Fallacy

Before comparing two options, strip them down.

Ask:

1. 1Does one option contain everything in the other plus an extra condition?
2. 2Can I draw the options as sets?
3. 3Is one set fully inside the other?
4. 4Would a frequency format make the relationship clearer?
5. 5Am I choosing the detailed option because it fits the story better?

If one statement is literally:

A and B

while the other is:

A

then A-and-B cannot have the larger probability under standard logic.

Frequently Asked Questions

Plausible Is Not the Same as Probable

A richer story can feel like a better prediction.

But probability follows set relationships, not narrative detail.

When one option contains an extra requirement, ask the simplest possible question:

Can the smaller, more specific group contain more cases than the larger group it belongs to?

Once you see the sets clearly, the conjunction rule becomes much harder to miss.