A bag, one hidden counter, and a draw that changes the odds
A bag contains a single counter, known to be either black or white with equal probability. You add one white counter, shake the bag, and draw out a counter at random — it's white. What is now the probability that the counter still left in the bag is also white?
Reveal the answer
2/3, not 1/2. Before drawing, there were two equally likely starting bags (black+white or white+white), giving three equally likely counters that could have been drawn, two of which are white and leave a white counter behind. Carroll included this among his trickiest 'Pillow Problems' precisely because most people's instinct says the odds should stay at 1/2 — an early, elegant demonstration of updating a probability on new evidence.
— Lewis Carroll, Curiosa Mathematica, Part II: Pillow Problems — 1893