Draw 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.
A counterfeit coin that might be heavy or light, a balance that only reports which side falls, and a hundred dollars a weighing. Counting rules out four; only a construction gets you five.
Una enfermedad que tiene una persona de cada doscientas, y una prueba sin ningún falso negativo. La respuesta refleja ante un positivo supera el noventa por ciento, y se equivoca por más de un factor de diez. Una multitud de mil personas lo enseña antes que el álgebra, y Bayes fija la cifra exacta en 100/1493.
Una acción con volatilidad cero, un call at the money, y la respuesta refleja — cero — que suspende entrevistas reales de trading. Un solo argumento de arbitraje la valora de tres maneras, y Black–Scholes lo confirma al final.
Coloca tres puntos al azar en un círculo y el triángulo contiene el centro exactamente una de cada cuatro veces. Dos demostraciones: un promedio honesto y luego dos monedas disfrazadas de geometría.