Bertrand's Box Paradox Calculator

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Classic uses the original three-box problem. Custom lets you explore any combination of box types.
After picking a box at random and drawing one coin at random, what colour did you see? The probability that the other coin matches is always higher than 1/2 in the classic setup.
P(other coin matches)Higher than intuition predicts
66.67%

Probability the remaining coin in the box is the same colour as the one you drew

Exact fraction2/3
P(other coin differs)33.33%
P(drawing that coin colour)50%
Favourable coin draws2
Total coins of that colour3
Total boxes3
Total coins6
66.67%
Below 1/3<34%Below 1/234%-50%Around 2/350%-67%Near certain67%+

The other coin is gold with probability 2/3 (66.7%), not the intuitive 50%.

  • You drew a gold coin. A same-colour box had twice as many gold coins to offer as a mixed box, so it is the more probable source of the coin in your hand.
  • Of the 3 gold coins across all boxes, 2 come from same-colour boxes (the Bayes numerator), giving 2/3.
  • Intuition anchors on "two possible boxes left, two outcomes" and answers 50%. Counting individual coins as equally likely - the correct approach - shifts the answer to 2/3.
  • This is structurally identical to the Monty Hall problem: new information (the colour of the drawn coin) updates probabilities asymmetrically, not simply by halving the remaining options.

Next stepTry switching to a custom setup and increasing the number of all-gold boxes - the probability climbs toward 100% as same-colour boxes dominate.

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