Bertrand's Box Paradox Calculator

Bertrand's Box Paradox is a classic conditional probability puzzle. In its standard form, three boxes each hold two coins. One box contains two gold coins, one contains a gold coin and a silver coin, and one contains two silver coins. A box is selected uniformly, one coin is drawn uniformly, and the observed coin is gold. The question asks for the probability that the unseen coin in the selected box is also gold. A quick but incorrect intuition says one-half because only the Gold-Gold and Gold-Silver boxes remain possible. The missing detail is that those two boxes are not equally likely after a gold coin has been observed. The Gold-Gold box has two faces that could produce the observation, while the Gold-Silver box has only one. Among the three equally likely gold-producing coin faces, two belong to the Gold-Gold box. Therefore the conditional probability that the other coin is gold is 2/3, or about 66.67%. The observation changes the relative likelihood of the boxes, which is the central lesson of the paradox. This calculator generalizes the puzzle by allowing multiple boxes of each type. If there are G Gold-Gold boxes and M Gold-Silver boxes, then there are 2G gold faces from the first group and M gold faces from the mixed group. Given that gold was observed, the probability of having selected a Gold-Gold box is 2G divided by 2G + M. Silver-Silver boxes do not appear in that formula because they cannot produce the stated gold observation, although they still belong to the experiment before any coin is seen. The puzzle is a compact demonstration of Bayes' theorem, selection effects, and the importance of counting outcomes under the evidence. It resembles medical-test interpretation, quality-control sampling, and other situations where an observation is more likely under one hidden cause than another. The answer depends on the sampling assumptions: boxes must be equally likely to be selected, and either coin in a chosen box must be equally likely to be drawn. If selection probabilities differ, those weights must be included explicitly. Use the custom counts to see how duplicated box types alter the posterior probability while preserving the same conditional reasoning.