Unit 1 of 4
Paradoxes That Fool You
- 23 chapters
- Video 1, chapters 1 to 23
This video traces famous paradoxes and surprising results to a hidden assumption in a picture, a probability model, a strategic choice or an infinite collection.
The first chapters check pictures and quantities against exact measurement, from the missing square to the missing dollar. Paradoxes of chance and averages, such as Monty Hall, the birthday paradox and Simpson's paradox, depend on how outcomes are chosen and how groups are weighted. In nontransitive dice, Penney's game and Braess's paradox, the better choice depends on what the others choose. The last chapters separate finite stages from limits and infinite sets, from Zeno's race and why 0.999... equals 1 to Hilbert's Hotel, Cantor's diagonal argument and the Banach-Tarski theorem.
Every chapter is part of one of 4 long videos. Each chapter page has the definitions and an example for every key term.
What the picture hides
The Missing Square
How the missing square puzzle, a dissection puzzle whose rearranged pieces seem to leave a hole, is explained by two unequal slopes, because the corners of its long edge are not collinear points.
The Coin Rotation Paradox
Why a coin rolling without slipping around an equal coin turns twice in the coin rotation paradox, explained by the circumference of its center's path, the turning line of centers and the frame of reference.
Aristotle's Wheel
How Aristotle's wheel paradox is resolved, why concentric circles fixed together share one angular speed, and why only the rim on the road rolls without slipping while the smaller circle combines rolling and sliding.
The Napkin Ring Paradox
Why napkin rings of equal height have equal volume whatever the size of the ball, explained through the napkin ring problem, cross-sections, the area of an annulus, Cavalieri's principle and the volume of a sphere.
The Potato Paradox
How a batch that dries from 99 percent to 98 percent water can lose half its mass, explained through the potato paradox, percentages of a changing total, percentage points and ratios.
The Missing Dollar
Where the missing dollar in the hotel bill goes, explained through the missing dollar riddle, double counting and net payment, with a ledger that follows every dollar to its final owner.
What random really means
Monty Hall
How the Monty Hall problem is settled by writing out the sample space under the host's rules, why the switching strategy wins with probability two thirds, and how conditional probability depends on what the host knows.
Bertrand's Box Paradox
How Bertrand's box paradox is resolved by counting equally likely outcomes of the selection procedure, and how Bayes' theorem shows that a gold coin makes the box with two gold coins twice as likely as the mixed box.
The Birthday Paradox
Why the birthday problem, also called the birthday paradox, gives a shared birthday among 23 people with probability about 50.7 percent, found from the number of pairs, the complement rule and the multiplication rule.
Simpson's Paradox
How Simpson's paradox lets one player lead on easy and on hard levels yet trail overall, why aggregated data give a weighted average of group rates, and what a confounding variable means for claims about cause.
The Friendship Paradox
How the friendship paradox compares the friend count of a random person with that of a random entry on all friend lists, and why size-biased sampling puts that average at or above the average degree of a social network.
The Waiting-Time Paradox
How the waiting-time paradox, a case of the inspection paradox, makes a passenger arriving at a random time wait longer than half the average gap between buses, explained by length-biased sampling and the expected waiting time.
Bertrand's Chord Paradox
Why three ways of choosing a random chord give one third, one half and one quarter in Bertrand's chord paradox, and why geometric probability needs a stated probability model.
When better goes in circles
Nontransitive Dice
How three fair nontransitive dice beat each other in a cycle, why their win probability relation is not a transitive relation, and why equal expected value does not decide a matchup.
Penney's Game
How the second player in Penney's game can pick a coin toss sequence that is more likely to appear first, why overlapping patterns decide the race, and how a state diagram gives the exact chances.
Braess's Paradox
How Braess's paradox lets a free shortcut slow every driver in a traffic network, explained by the Nash equilibrium of selfish routing and how it differs from the social optimum.
When infinity changes the rules
Zeno's Achilles and the Tortoise
How the race of Achilles and the tortoise, one of Zeno's paradoxes, is resolved by a geometric series, a convergent series whose infinitely many stage times add up to the catch-up time given by relative speed.
Why 0.999... Equals 1
Why the repeating decimal 0.999... names exactly the real number 1, explained by the limit of a sequence of finite truncations whose gaps from 1 become smaller than any positive number.
Hilbert's Hotel
How Hilbert's Hotel, full yet able to take new guests, works through one-to-one correspondences, why a countably infinite set matches a proper subset of itself, and how a pairing function seats infinitely many buses.
Can Every Fraction Be Counted?
How the rational numbers, a dense set on the number line, still form a countable set, through a diagonal enumeration of a fraction grid that keeps each fraction only in lowest terms.
Cantor's Diagonal Argument
How Cantor's diagonal argument, a proof by contradiction built from decimal expansions, shows that the real numbers between 0 and 1 form an uncountable set that no numbered list can contain.
The Ross-Littlewood Urn
How the Ross-Littlewood paradox, a supertask that adds ten balls and removes one at every stage, ends with an empty set-theoretic limit while its counts form a divergent sequence.
Banach-Tarski and What the Theorem Actually Says
What the Banach-Tarski paradox says about splitting a solid ball into finitely many sets and moving them by rigid motions into two balls, and how nonmeasurable sets, the axiom of choice and cardinality explain it.