Conditioning, expectation, distributions and the paradoxes that make strong candidates answer too fast.
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.
A disease one person in two hundred carries, and a test with no false negatives at all. The reflex answer to a positive result is above ninety percent, and it is wrong by more than a factor of ten. A crowd of a thousand people shows why before the algebra does, and Bayes puts the exact figure at 100/1493.
Drop three random points on a circle and the triangle contains the center exactly a quarter of the time. Two proofs: an honest average, then two coins wearing a geometry costume.