The base rate fallacy happens when you judge a probability from specific evidence—a test result, witness report, or alert—while ignoring or underweighting how common the thing is in the first place. The fastest cure is seeing the arithmetic fail a few times, so this article works through seven examples. Each gives you a base rate, evidence rates, a natural-frequency table, and the correct answer.
The two ingredients (and two distinct mistakes)
Every problem below has the same skeleton: a base rate (how common the condition is, also called prevalence or prior probability) and evidence rates (such as sensitivity and false-positive rate). A vague claim that a test is “99% accurate” may be insufficient because it does not necessarily tell you how the test performs on each group.
Two related mistakes can appear:
- Base-rate neglect: you ignore or underweight the prior probability while focusing on vivid evidence. This distorts the result, but it does not necessarily mean assuming a 50/50 prior.
- Conditional-probability reversal: you confuse P(positive | sick) with P(sick | positive). The second probability also depends on the base rate.
These errors often occur together, but neither is simply the cause or mechanism of the other. Now the examples.
Example 1: The rare-disease test
Setup: Prevalence is 1 in 1,000. Sensitivity is 99%, and the false-positive rate is 5%. You test positive. How likely are you to be sick?
Take 100,000 people:
| Group | Count | Test positive |
|---|---|---|
| Sick | 100 | 99 |
| Healthy | 99,900 | 4,995 |
P(sick | positive) = 99 ÷ (99 + 4,995) ≈ 1.9%.
The intuitive answer—“near 99%”—confuses sensitivity with the probability of illness after a positive result. False positives from the much larger healthy group greatly outnumber true positives.
Example 2: The taxi cab witness
Adapted from a problem studied by Kahneman and Tversky. Setup: 85% of a city’s cabs are Green and 15% are Blue. A witness says a hit-and-run cab was Blue. Under similar conditions, the witness identifies either color correctly 80% of the time.
Out of 100 cabs:
| Group | Count | Witness says “Blue” |
|---|---|---|
| Blue | 15 | 12 |
| Green | 85 | 17 |
P(Blue | witness says Blue) = 12 ÷ (12 + 17) ≈ 41%.
The testimony is meaningful: it raises the probability from 15% to 41%. But it does not make Blue more likely than not. An argument that substitutes the witness’s 80% accuracy for the probability that this cab was Blue neglects the fleet’s composition.
Example 3: The engineer–lawyer description
Adapted from Kahneman and Tversky’s engineer–lawyer problem. Setup: A group contains 70 lawyers and 30 engineers. One randomly selected person is described as quiet, fond of puzzles, and good with tools. For this labeled hypothetical, suppose 60% of engineers and 20% of lawyers fit that description.
| Group | Count | Fit the description |
|---|---|---|
| Engineers | 30 | 18 |
| Lawyers | 70 | 14 |
P(engineer | description) = 18 ÷ (18 + 14) = 56.25%.
The description is three times as likely among engineers, yet the answer is only modestly above 50% because lawyers are more common in the starting group. A representative-sounding description does not erase the prior distribution.
Example 4: The spam filter
Setup (hypothetical): In a corporate mail stream, 2% of messages are spam. A filter flags 98% of spam and 2% of legitimate mail. A message is flagged. How likely is it to be spam?
Take 10,000 messages:
| Group | Count | Flagged |
|---|---|---|
| Spam | 200 | 196 |
| Legitimate | 9,800 | 196 |
P(spam | flagged) = 196 ÷ (196 + 196) = 50%.
Even with strong detection, this particular combination of prevalence and false-positive rate makes half the flagged messages legitimate. Rare targets can produce many false alarms unless the detector’s false-positive rate is sufficiently low relative to the base rate. The same reasoning can apply to fraud alerts and security monitoring.
Example 5: Mass screening for a very rare trait
Setup (hypothetical): A trait is present in 1 out of 100,000 people. A screen has 99% sensitivity and a 1% false-positive rate.
Take 100,000 people:
| Group | Count | Test positive |
|---|---|---|
| Has the trait | 1 | 0.99 |
| Does not have it | 99,999 | 999.99 |
P(trait | positive) = 0.99 ÷ (0.99 + 999.99) ≈ 0.1%.
Fractional expected counts are acceptable here because the table represents an average across repeated populations. The result is not a general verdict on screening. It shows only that, under these hypothetical rates, an initial positive would provide limited evidence and would need further evaluation before supporting a conclusion.
Example 6: The airport scanner alert
Setup (hypothetical): Suppose 0.5% of screened bags contain a prohibited item. A scanner flags 90% of bags containing one and falsely flags 4% of other bags. A bag triggers an alert. How likely is it to contain a prohibited item?
Take 10,000 bags:
| Group | Count | Flagged |
|---|---|---|
| Contains prohibited item | 50 | 45 |
| Does not contain one | 9,950 | 398 |
P(item | flagged) = 45 ÷ (45 + 398) ≈ 10.2%.
The alert raises the probability from 0.5% to about 10.2%, so it is informative. But most alerts are still false positives because ordinary bags are vastly more common. Treating the 90% detection rate as a 90% probability that the flagged bag contains an item reverses the conditional probability.
Example 7: The workplace drug test (LSAT-flavored)
Setup (hypothetical): An employer argues, “Our test identifies users 95% of the time, so employees who test positive should be presumed users.” Suppose 1% of employees use the substance, sensitivity is 95%, and the false-positive rate is 5%.
Take 10,000 employees:
| Group | Count | Test positive |
|---|---|---|
| Users | 100 | 95 |
| Non-users | 9,900 | 495 |
P(user | positive) = 95 ÷ (95 + 495) ≈ 16%.
The employer treats one performance measure as if it settled the probability of the conclusion. An LSAT flaw answer might describe this as failing to consider how frequently the condition occurs in the relevant population.
The four-step method
You do not need to write Bayes’ theorem under time pressure. Build a frequency table:
- Pick a population that makes the base rate easy to represent.
- Split it by the base rate into “has it” and “does not.”
- Apply the evidence rates to both groups.
- Divide: true positives ÷ all positives.
The same method works with tests, witnesses, filters, and alerts.
Where this shows up on the LSAT and GRE
On LSAT Logical Reasoning, watch for arguments moving from “the method detects most genuine cases” to “this detected case is probably genuine.” Also watch for raw-count comparisons that omit relevant denominators. Practice recognizing those structures in PurrLearn’s LSAT Logical Reasoning flaw quiz.
The GRE’s Analyze an Argument essay was retired when the shortened GRE launched in September 2023. It is therefore not a current GRE task. Older Argument prompts can still provide general reasoning practice, including practice identifying missing prevalence data, but they should not be mistaken for the present test format.
For more interactive probability and selection-bias practice, try the Monty Hall problem quiz and survivorship-bias quiz. They address related habits: tracking the full set of possibilities and asking which cases the available evidence leaves out.
FAQ
What is a simple base rate fallacy example?
A disease affects 1 in 1,000 people; a test has 99% sensitivity and a 5% false-positive rate; you test positive and assume you are almost certainly sick. The probability under those assumptions is about 1.9% because false positives outnumber true positives.
Is the base rate fallacy the same as confusing P(A|B) with P(B|A)?
No. Conditional-probability reversal swaps two different probabilities. Base-rate neglect means ignoring or underweighting prior frequency. They can occur separately, although a mistaken answer may involve both.
How does the base rate fallacy appear on the LSAT?
An argument may conclude that an identified case is probably genuine because the method detects genuine cases reliably, without considering how rare genuine cases are or how often false positives occur.
How do I avoid the base rate fallacy?
Before reacting to a result, alert, or match, ask how common the target condition is. Then place the base rate, sensitivity, and false-positive rate into a frequency table. If the base rate is unknown, the evidence may still matter, but you cannot calculate the posterior probability from the detection rate alone.
Conclusion
These seven examples share one structure: a prior probability, evidence rates for each group, and a question about probability after observing the evidence. The fix is mechanical—pick a population, split it by the base rate, apply the evidence rates, and divide. Practicing that sequence makes it easier to spot the trap in LSAT arguments, older GRE reasoning material, and everyday probability claims.