Going in and winning are different events: the short shot clears two hurdles and wins 0.35 of the time against the long shot's 0.40. The article prices how wrong the reflex is in two currencies, a break-even overtime rate of 4/7 and a break-even make rate of 80 percent at a coin-flip overtime. It also names the objective under which the reflex is right, since the short shot scores 1.40 expected points against 1.20 and still wins fewer games.
Four over fifty-two squared is exactly right for the question where the first card goes back, which is what makes it hard to catch. Removing a king shrinks the numerator proportionally more than the denominator, and the counting route through 1326 two-card hands confirms one in 221 without mentioning order at all.
Three and a half is the exact average of a plain die, which is why it survives being double-checked. The rule does not reweight six outcomes, it deletes one, leaving a uniform payoff on five faces and an answer of four. The procedure costs 1.2 rolls on average, and the version where the reroll is your choice is a different game worth 4.25.
Deux cassures au hasard, trois morceaux, et trois inégalités qui n'en font qu'une. Le quart tombe d'un carré sans la moindre intégrale, et le morceau le plus long, 11/18 en moyenne, explique pourquoi la réponse paraît trop basse sans l'être.
Deux portes restantes ne font pas deux portes égales : votre premier choix est resté figé à 1/3, et les 2/3 restants se sont entassés sur la seule porte encore fermée. Le nombre ne dit rien des portes, il dit tout du présentateur. Laissez-le ouvrir au hasard, montrez la même chèvre, et changer ne vaut plus que 1/2.
Told that one of two children is a girl, the chance both are girls is 1/3. Watch a girl open the door instead and it is 1/2, from the same four families and the same prior. One likelihood separates them: a mixed family always satisfies the statement, but sends the girl to the door only half the time. Push the identifying detail to a girl born on a Tuesday and the answer slides to 13/27.
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.
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.