38 Hempel’s Raven
Scientific hypotheses are usually universal claims—for example, “All ravens are black.” By standard logic that statement is equivalent to its contrapositive “All non-black things are non-ravens.” Hempel pointed out a tension between that equivalence and ordinary confirmation. A sighting of a black raven clearly supports the hypothesis, yet the contrapositive implies that inspecting a green apple also supports it, because the apple is a non-black non-raven. People feel the apple provides no real evidence about raven plumage. Bayesian resolution shows why the intuition and the logic can both be right: a green apple does raise the probability, but only by an imperceptibly tiny amount once one factors in the overwhelming prior number of non-black objects. Hempel’s puzzle therefore sharpens the distinction between formal confirmation (any logically relevant instance) and informativeness (evidence that changes probabilities appreciably).
39 Unexpected Hanging
A judge announces to a prisoner, “You will be hanged next week, but the day will be a surprise.” The prisoner reasons backwards: the hanging cannot occur on Friday—he would know by Thursday night. Once Friday is excluded, Thursday becomes predictable and is likewise ruled out, then Wednesday, and so on, proving to the prisoner that execution is impossible. Yet he is hanged on Wednesday, entirely surprised. The paradox turns on self-referential reasoning about future knowledge. Formal epistemic logics show that embedding statements about what will be known within a public announcement can destroy the fixed-point needed for consistent belief, so the judge’s declaration is not a straightforward promise but an unstable loop in the agent’s knowledge state.
40 Sleeping Beauty
Beauty is put to sleep on Sunday. A fair coin is tossed. Heads: she is awakened on Monday, then released. Tails: she is awakened Monday and Tuesday, with memory erased after Monday. Each time she awakens she is asked, “What is the probability the coin landed heads?” The Halfer camp says ½, matching the physical coin. The Thirder camp reasons that awakenings are in a 1 : 2 ratio (one under heads, two under tails), so heads should be credited only 1/3. The dispute exposes a gap between objective chance and self-locating probability. Modern treatments employ event-splitting frameworks: if Beauty conditions on “today is an awakening” rather than “the coin came up heads,” the 1/3 answer follows; if she conditions on the external random process alone, 1/2 follows. The problem therefore illustrates how reference classes must be fixed before updating.
41 Two-Envelope
There are two envelopes, one with twice the amount as the other envelope. You open one envelope and find $X. Logically, the other envelope contains either 2 X or X⁄2, but not both. The seductive calculation “expected value = ½·2X + ½·X⁄2 = 1.25 X” silently treats X as both the high and the low amount. A correct analysis conditions on the underlying pair (A, 2A): once X is observed the chance that you hold the smaller amount is not ½ unless you already know the distribution of A. Under common symmetric priors the expectation of switching equals X, erasing any universal advantage. The paradox demonstrates that unconditional symmetry cannot be projected onto the observed amount without double-counting possibilities.
42 Absent-Minded Driver
A driver trying to reach home must decide at each motorway exit whether to leave, but suffers amnesia and never knows which exit he is at. Pay-offs: first exit = 2, second exit (home) = 4, missing both = 0. If he pre-commits to “always exit,” he guarantees 2; “never exit” guarantees 0. Surprisingly, assigning a mixed strategy—exit with probability p at any junction—can yield an expected value > 2 (with p≈⅓). The puzzle shows that behavioural policies rather than state-contingent choices are rational when the agent’s information is systematically limited; it also motivates the study of imperfect recall in extensive-form games.
43 St Petersburg Game
A fair coin is tossed until it shows heads; the pot starts at $2 and doubles after each tails. The expected monetary value diverges: ∑ 2ⁿ⁻¹⁄2ⁿ = ∞. Yet few people will pay even $25 to play. The contradiction between infinite expected value and modest reservation price fueled Daniel Bernoulli’s conception of diminishing marginal utility—each additional dollar adds less subjective value. Modern treatments use risk aversion, bounded rationality, or finite bankroll constraints to align intuitive valuations with utility-theoretic expectations.
44 Monty Hall
After the contestant picks one of three doors, the host—knowing all contents—always opens a different door to reveal a goat, then offers a switch. Because the host’s action is correlated with the initial hidden allocation, probability mass (2⁄3) transfers to the single unopened alternative. Experimental replications and Bayesian trees demonstrate the switch advantage, yet persistent mis-intuition highlights that human reasoning defaults to equiprobability once an option is visibly removed, neglecting conditional information embedded in the host’s choice.
45 Bertrand’s Chord
“What fraction of random chords in a circle are longer than a side of the inscribed equilateral triangle?” Method 1: pick two random perimeter points ⇒ 1⁄3. Method 2: pick a random radius and a point on it ⇒ 1⁄2. Method 3: pick a random interior point as chord midpoint ⇒ 1⁄4. Because “random chord” is underspecified, different sampling protocols yield different answers. Bertrand’s example warns that geometric probability requires an explicit symmetry or selection rule—uniformity over one parameter does not translate into uniformity over another.
46 Simpson’s Paradox
A medical treatment improves recovery for men and for women considered separately (say 70 % vs 60 % in each group) yet shows a lower overall recovery rate when the data are combined. The reversal emerges when gender also influences admission severity: the treatment group may include disproportionately serious cases. Simpson’s paradox underlines that aggregated proportions can mislead whenever a lurking variable affects both the independent and the dependent factors. Causal diagrams and stratified analysis are the modern safeguards.
47 Multiple-Comparison (Gelman) Paradox
If a study probes 100 gene–disease links at α = 0.05, roughly five “significant” findings will appear by chance alone. Selective publication of those five inflates false discovery rates while each p-value still reads < 0.05. The paradox illustrates that significance thresholds lose meaning without accounting for the search process—the so-called garden of forking paths. Hierarchical modelling and false-discovery-rate controls recalibrate evidence to the full experiment space.
48 Prosecutor’s Fallacy
A DNA match with random-match probability 1 : 1 000 000 is presented as “only a one-in-a-million chance the defendant is innocent.” Yet if ten million people could have left the sample, roughly ten innocents would also match. The fallacy confuses P(evidence | innocent) with P(innocent | evidence), ignoring base rates. Bayesian updating with population priors converts a dramatic-sounding rarity into a much more modest posterior certainty, reminding courts to distinguish conditional directions.
49 Observation-Selection / Anthropic
Why do we observe physical constants finely tuned for life? Because only a universe (or cosmic region) with such constants permits observers. The reasoning seems circular—yet in a multiverse setting it yields testable predictions about probability distributions conditional on observer existence. The paradox forces philosophers to decide whether self-sampling assumptions are legitimate evidence or merely restatements of survival bias.
50 Fermi Paradox
Simple astrophysical estimates suggest thousands of civilizations should have arisen in the Milky Way, yet radio telescopes hear only silence. Proposed resolutions span (i) life is extremely rare, (ii) civilizations self-destruct, (iii) they hide, (iv) we are early, (v) detection methods are naive. The paradox is pivotal to astrobiology: it transforms a vague feeling of loneliness into a quantitative problem about birth rates, expansion speeds, and technological signatures.
51 Olbers’ Paradox
In an infinite, eternal, static universe filled evenly with stars, every line of sight ends on a stellar surface, so the night sky should be ablaze. The darkness overhead implies at least one premise is wrong. Modern cosmology rejects all three: finite stellar lifetimes, finite cosmic age, and cosmic expansion together thin and red-shift starlight, reconciling observation with theory and turning a 19th-century puzzle into an early hint of the Big Bang.
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