15 Probability Pitfalls Visualized: Why 99% Accurate Tests Can Be 91% Wrong

Model Perspective · Illustrated Models Issue 1

A test with 99% accuracy. You test positive. What's the probability you actually have the disease?

Most people say 99%. If the disease prevalence is 0.1%, the answer is 9%.

The intuition is off by a factor of ten. The calculation isn't hard; the problem is that intuition gives a confident answer so you never think to calculate.

Below are 15 of the most common probability pitfalls, each with a diagram. They fall into three categories:

Wrong denominator (01–04)

Sampled data (05–09)

Mistaking luck for a pattern (10–15)

A summary diagram at the end can be saved for reference.

01 Category 1: Wrong Denominator

01 Base Rate Fallacy: Positive ≠ Sick

100,000 tested, 1,098 positive, only 99 truly sick

Test 100,000 people. Prevalence is 0.1%, so 100 are truly sick; the test catches 99 of them (99% sensitivity).

The remaining 99,900 healthy people generate 1% false positives = 999 false positives.

Total positives = 1,098; true positives = 99 ≈ 9%.

"99% accurate" means 99% of sick people test positive. You want to know: of those who test positive, how many are sick? The denominators differ. The rarer the disease, the more healthy people get mislabeled, and the less a positive result means.

That's why doctors retest. The first positive raises probability from 0.1% to 9%; a second positive on that basis pushes it to ~90%.

02 Prosecutor's Fallacy: "One in a Million" ≠ Probability of Innocence

DNA alone puts guilt probability closer to 1/10

Crime-scene DNA matches the suspect. Expert says "random match probability is one in a million." It's tempting to think the suspect's innocence probability is also one in a million.

Change the denominator. In a city of 10 million, one in a million means ~10 people match. With no other evidence, the suspect is just one of those 10 → guilt probability ≈ 1/10.

This is the same error as the base rate fallacy: confusing "probability an innocent person matches" with "probability a matching person is innocent." Courts have made this mistake often enough to earn a named fallacy.

03 Simpson's Paradox: Every Group Wins, Combined Loses

A got mostly hard-to-treat large stones

Real data from a 1986 BMJ study comparing two kidney-stone treatments:

Small stones: A 93% success, B 87%

Large stones: A 73%, B 69%

Combined: A 78%, B 83%

Each subgroup favors A, but overall B wins.

Patients weren't randomized. Large stones are harder; doctors preferred A (open surgery) for them, so A handled 263 large stones vs. B's 80. A's overall rate was dragged down by the tougher cases.

Common in education too: a teacher's low class average may simply reflect a weaker cohort.

04 Conjunction Fallacy: Each Step 95%, Whole Project Only 60%

Longer plans make 'all smooth' harder

A project has 10 steps, each 95% likely to succeed. Overall success = 0.95^10 ≈ 60%. If each step is 90%, 10 steps yield only 35%.

When estimating a whole project, people picture each step as "pretty stable" and give the whole project a high score. This substitutes the per-step denominator for the whole-project denominator. Renovations, moves, study-abroad plans—more steps mean more overruns for this reason.

02 Category 2: Sampled Data

05 Survivorship Bias: Reinforce Where the Bullet Holes Aren't

Planes hit in the engine didn't return

WWII returning planes had bullet holes clustered on wings and fuselage, few near engines. Intuition: armor where holes are dense.

Statistician Abraham Wald argued the opposite: engines had no holes because planes hit there never returned. Armor where the returning planes have *no* holes.

Wald's original report was more complex, but the core message: the sample you see has been filtered by a "survival" rule.

"Dropouts include many billionaires" and "old houses are sturdy" work the same way. The failed dropouts and collapsed houses are invisible.

06 Inspection Paradox: Average Class 25, Students Feel 52

Average by class vs. average by student, off by 2x

A school has 4 classes: three small (10 each) and one large (70). School's average class size = 25.

Ask students "how big is your class?" 100 students total, 70 are in the large class. Student-reported average = 52, double the school's figure.

Both calculations are correct; denominators differ: one averages over classes, the other over students. You're more likely to be in a larger group.

Everyday examples:

Your bus wait often falls in a longer interval because long intervals occupy more time.

Gyms and restaurants feel crowded because you're there when they're crowded.

07 Berkson's Paradox: "Good-Looking People Have Bad Personalities"?

After filtering, unrelated variables become negatively correlated

Assume looks and personality are independent. You consider someone only if "looks + personality" exceeds a threshold.

Simulation result: in the full population correlation ≈ 0; in the considered subset correlation = -0.67.

Average-looking people who make the cut must have great personalities; stunning people need only average personalities. After filtering, it looks like a trade-off.

"Elite students lack social skills" and "tasty restaurants have bad service" are often the same artifact.

08 Friendship Paradox: Your Friends Have More Friends Than You

Popular people get counted many times

In a simulated 2,000-person network, average friends = 6. But the average friend-of-a-friend has 15.9 friends. 89% of people have fewer friends than their friends do.

Popular people appear on many friend lists; less popular people appear on few. When counting "friends' friend counts," popular people are over-represented.

Sociologist Scott Feld published this in 1991. A comforting corollary: feeling everyone else is more social is likely just this algorithmic illusion.

Practical use: early epidemic detection—ask a random person to name a friend, then test that friend. Friends have wider networks and get infected earlier.

09 Average Trap: Most People Earn Below Average Wage

A few high incomes pull the average right

Income distribution has a long right tail: most cluster in the lower-middle, a few earn very high.

In the illustrative data, median = 5,000, mean = 6,390, 64% below mean. Feeling "averaged" when seeing average wage is statistically justified.

The mean is pulled right by extreme values. For income, housing prices, wait times—any long-tailed metric—the median better represents a typical person.

03 Category 3: Mistaking Luck for a Pattern

10 Regression to the Mean: Scored First, Next Time Likely Worse

Skill unchanged, luck returned to normal

Psychologist Daniel Kahneman told flight instructors that praise works better than criticism. An instructor objected: praising a good flight led to a worse next flight; criticizing a bad flight led to improvement.

Praise and criticism likely had no effect. Performance = skill + luck. An exceptionally good outcome usually includes exceptionally good luck; next time luck reverts to average, so performance drops.

Simulation of 100,000 people taking two tests: top 10% on first test averaged 2.48 SD above mean; on second test they fell to 1.23 SD. 84% declined, yet their skill never changed.

Sports "cover jinx" (athletes decline after magazine cover) and "new leader quick wins" (often appointed at a low point) are largely regression to the mean.

11 Gambler's Fallacy: 5 Reds in a Row, Black Is Due?

Roulette has no memory

After 5 reds, probability of black on next spin remains 18/37 ≈ 48.6%.

People think "black is due" because they interpret "long-run red/black balance" as "short-run correction." The law of large numbers works by diluting early deviations with massive new trials, not by cancelling them.

12 Streak Illusion: 20 Coin Flips, Nearly Half Show a Run of 5

True randomness naturally clusters

Ask people to write a "random" 20-flip sequence; most alternate too regularly, rarely writing 4-5 same faces in a row.

Real 20 flips: longest run averages 4.7; probability of a run ≥5 is 46%.

True randomness clusters; human-generated "randomness" is too even. When you see a winning streak or a string of limit-up stocks, first check how long a pure-random streak would be.

Twist: a famous 1985 study claimed basketball's "hot hand" is an illusion; a 2018 correction (Miller & Sanjurjo) found a statistical bias in that study. After correction, the hot hand may be real but much weaker than fans believe.

13 Law of Small Numbers: Highest and Lowest Rates Are Both Small Counties

Smaller sample, larger fluctuation

From Thinking, Fast and Slow : U.S. counties with lowest kidney-cancer rates are mostly rural, sparse counties. Easy to invent reasons: clean air, healthy diet. But counties with highest rates are *also* mostly rural, sparse counties.

Simulation: 1,000 counties, true rate identical (10 per 10,000). Small counties (few thousand people) show rates from 0 to 30+; large counties (hundreds of thousands) cluster tightly around 10.

Small samples fluctuate wildly, so "top/bottom" lists are dominated by small samples. "Best schools," "top-performing funds," "most dangerous cities"—check sample size before trusting the ranking.

14 Birthday Problem: 23 People, >50% Chance of Shared Birthday

23 people form 253 pairs

In a group of 23, probability of at least one shared birthday = 50.7%; with 50 people, 97%.

Intuition underestimates because you think "someone shares *my* birthday." The question asks "any pair shares a birthday." 23 people create 253 pairs, each a chance.

Many "what a coincidence" moments are driven by the number of opportunities, not the probability per opportunity.

15 Law of Truly Large Numbers: 1-in-10,000 Daily Event → 84% Lifetime Chance

Rare × trials = not rare

An event with daily probability 1/10,000. Over 50 years (≈18,250 days), probability of at least one occurrence = 84%.

Conversely, a one-in-a-million coincidence for an individual happens to ~1,400 people daily in a population of 1.4 billion. Statisticians Diaconis and Mosteller call this the "law of truly large numbers": with a large enough sample, any outrageous thing will happen.

Personal relevance: driving, using electricity, climbing—single-event risk is tiny, but a lifetime involves tens of thousands of repetitions.

04 One Diagram to Take Away

15 probability pitfalls in one diagram

Take away one sentence: when a probability surprises you, ask three questions: Who is the denominator? Who filtered the sample? Would pure randomness produce this too?

These three questions won't catch every error, but they block most of the 15 above.

References Charig C R, Webb D R, Payne S R, Wickham J E. Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy. BMJ , 1986, 292: 879–882. Thompson W C, Schumann E L. Interpretation of statistical evidence in criminal trials: The prosecutor's fallacy and the defense attorney's fallacy. Law and Human Behavior , 1987, 11(3): 167–187. Tversky A, Kahneman D. Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review , 1983, 90(4): 293–315. Mangel M, Samaniego F J. Abraham Wald's work on aircraft survivability. Journal of the American Statistical Association , 1984, 79(386): 259–267. Feld S L, Grofman B. Variation in class size, the class size paradox, and some consequences for students. Research in Higher Education , 1977, 6: 215–222. Berkson J. Limitations of the application of fourfold table analysis to hospital data. Biometrics Bulletin , 1946, 2(3): 47–53. Feld S L. Why your friends have more friends than you do. American Journal of Sociology , 1991, 96(6): 1464–1477. Daniel Kahneman. Thinking, Fast and Slow . Translated by Hu Xiaojuan et al. CITIC Press, 2012. Gilovich T, Vallone R, Tversky A. The hot hand in basketball: On the misperception of random sequences. Cognitive Psychology , 1985, 17(3): 295–314. Miller J B, Sanjurjo A. Surprised by the hot hand fallacy? A truth in small numbers. Econometrica , 2018, 86(6): 2019–2047. Diaconis P, Mosteller F. Methods for studying coincidences. Journal of the American Statistical Association , 1989, 84(408): 853–861. Simulations in this article (Berkson, regression to mean, friendship paradox, streaks, small-county rates, income distribution) were programmed by the author; parameters are illustrative.

Code example

, 1991, 96(6): 1464–1477.
[8] 丹尼尔·卡尼曼. 思考，快与慢. 胡晓姣等译. 中信出版社, 2012.
[9] Gilovich T, Vallone R, Tversky A. The hot hand in basketball: On the misperception of random sequences.