Conditionnement, espérance, lois, et les paradoxes qui font répondre trop vite les bons candidats.
6 questionsDraw one coin from a thousand, flip ten heads, and the chance it is the two-headed one is 0.5062. Both reflex answers miss, in opposite directions: ninety-nine percent ignores the bag, one in a thousand ignores the flips. Counting patterns gets the exact figure with no Bayes notation at all, and the reason it lands on a coin flip is that 2^10 happens to sit next to the size of the bag.
Fifty white marbles, fifty black, two jars, and a fair coin choosing which jar gets drawn from. The even split gives exactly one half, and so does every other split where the jars are the same size. Isolating a single white marble reaches 74/99, an exchange argument proves nothing beats it, and three quarters turns out to be a ceiling no arrangement ever touches.
Drop chips at random into dough, cut it into a hundred cookies, and ask how many chips guarantee no bare cookie nine times out of ten. Five hundred chips, five per cookie on average, works about half the time. Inclusion-exclusion pins the answer at 683, a closed form you can solve on a whiteboard agrees, and the coupon collector's mean of 518.7 is the sophisticated wrong answer.
Six slots in a ring, two of them marked side by side. You land on a blank one and get one move: step forward, or draw a fresh slot at random. Both look like two in six. Stepping is one in four, drawing again is one in three, and the whole gap comes from the fact that the two marks are touching. Pull them apart and the advice reverses.
Une maladie que porte une personne sur deux cents, et un test sans le moindre faux négatif. La réponse réflexe à un résultat positif dépasse les quatre-vingt-dix pour cent, et elle se trompe d'un facteur dix. Une foule de mille personnes montre pourquoi avant que l'algèbre ne s'en mêle, et Bayes fixe la valeur exacte à 100/1493.
Placez trois points au hasard sur un cercle et le triangle contient le centre exactement une fois sur quatre. Deux preuves : une moyenne honnête, puis deux pièces déguisées en géométrie.