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Classic puzzles and the logic that makes them work.
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A standard sudoku grid is what size?
Chance, Odds and Bad Intuition
Human intuition about probability is not merely imprecise. It is wrong in specific, repeatable directions, and those directions have names.
About this topic
Reasoning about chance is a late arrival in human thought. Serious probability theory dates only from correspondence between Pascal and Fermat in the 1650s, which is remarkable given how long people had been gambling. The delay is a clue: the subject is genuinely counter-intuitive, and the errors it invites are not random but systematic, appearing in the same shape in person after person. Some of those errors are about memory of the past — expecting a coin to correct itself, reading a run as evidence of a pattern. Others are about scale, like underestimating how quickly coincidences become likely in a large group, or forgetting that a rare condition makes even an accurate test mostly wrong when it fires. The puzzles below are the classic demonstrations, each chosen because the correct answer still feels wrong after it is explained.
Questions & answers
Chance, Odds and Bad Intuition: all 10 questions and answers
Every question in this quiz is listed below with its correct answer and the reasoning behind it. Play first if you would rather not see the answers — or read through as a study sheet.
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1The gambler's fallacy is the belief that what?
- A Long odds always pay better than short ones
- B A run of one outcome makes the opposite outcome more likely on the next independent trial Correct
- C Any sequence of results must eventually repeat itself
- D Skilled players can influence a random device
Answer: B. A run of one outcome makes the opposite outcome more likely on the next independent trial
Why: A fair coin has no memory of what it has just done. The related hot-hand belief runs the other way, treating a streak as evidence that it will continue.
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2In the birthday problem, roughly how many people are needed before a shared birthday is more likely than not?
- A About 23 Correct
- B About 60
- C About 128
- D About 183
Answer: A. About 23
Why: The count that matters is pairs, not people, and 23 people make 253 possible pairings. Reaching near-certainty takes only about 70.
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3The conjunction fallacy occurs when someone judges what?
- A A common event to be rarer than it is
- B Two independent events to be dependent
- C A long sequence to be unrepeatable
- D A detailed combination of events to be more likely than one of its parts alone Correct
Answer: D. A detailed combination of events to be more likely than one of its parts alone
Why: Two conditions can never be more probable than either condition on its own. The detail makes the story feel more plausible even as it makes the claim mathematically narrower.
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4Why does a highly accurate test for a rare condition still produce mostly false alarms?
- A Because rare conditions cannot be detected reliably
- B Because errors accumulate as more people are tested
- C Because the far larger group without the condition generates more false positives than the small group generates true ones Correct
- D Because accurate tests are calibrated for common conditions
Answer: C. Because the far larger group without the condition generates more false positives than the small group generates true ones
Why: This is base rate neglect, and it survives explanation remarkably well. Doctors and statisticians have both been shown to misjudge the same problem when it is worded as a story rather than as frequencies.
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5Simpson's paradox occurs when what happens?
- A An average is dragged by a single extreme value
- B A trend that appears in every subgroup reverses when the groups are combined Correct
- C Two variables correlate with no causal link
- D A sample is too small to draw conclusions from
Answer: B. A trend that appears in every subgroup reverses when the groups are combined
Why: It arises when group sizes differ and the grouping variable itself influences the outcome. A well-known example involved graduate admissions data that looked biased in aggregate and not within any single department.
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6The expected value of a bet is calculated how?
- A Each outcome multiplied by its probability, then all of those added together Correct
- B The largest possible win divided by the stake
- C The most frequent outcome across many trials
- D The chance of winning minus the chance of losing
Answer: A. Each outcome multiplied by its probability, then all of those added together
Why: The result need not be an outcome that can actually occur, which is why a die has an expected value of 3.5. It describes the long-run average rather than any single throw.
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7Regression to the mean describes what?
- A Data drifting steadily upward over time
- B Errors cancelling out within a single reading
- C A sample becoming more varied as it grows
- D Extreme measurements tending to be followed by ones closer to the average Correct
Answer: D. Extreme measurements tending to be followed by ones closer to the average
Why: Francis Galton noticed it in the heights of parents and children in the 1880s. It quietly manufactures the impression that criticism works and praise backfires, since both usually follow an extreme performance.
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8The law of large numbers states what?
- A Rare events become impossible in large samples
- B Probabilities change as trials accumulate
- C The average of many independent trials converges toward the true expected value Correct
- D Large samples always contain outliers
Answer: C. The average of many independent trials converges toward the true expected value
Why: It says nothing about any individual trial or short run. Misreading it as a promise that things will even out shortly is exactly the gambler's fallacy.
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9Survivorship bias arises when what happens?
- A Participants withdraw from a study at random
- B Only the cases that made it through are examined, while the rest are invisible Correct
- C Older records are given too much weight
- D Two groups are compared without matching their ages
Answer: B. Only the cases that made it through are examined, while the rest are invisible
Why: The classic case is wartime aircraft: armour was nearly added where returning planes showed bullet holes. Abraham Wald pointed out that those were precisely the hits a plane could survive.
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10Two events are described as independent when what is true?
- A The occurrence of one leaves the probability of the other unchanged Correct
- B They cannot both happen at once
- C They occur equally often
- D They share the same cause
Answer: A. The occurrence of one leaves the probability of the other unchanged
Why: Mutually exclusive events are a different idea entirely and are strongly dependent, since one happening rules the other out. Confusing the two terms is a common source of error in probability problems.
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