Bertrand's Paradox Calculator

This version of Bertrand's paradox is the famous three-box conditional probability problem, also called Bertrand's box paradox. Three visually identical boxes contain different pairs of coins: two gold coins in the first, one gold and one silver coin in the second, and two silver coins in the third. A box is selected at random and one coin is drawn at random. Once the drawn coin is observed to be gold, the problem asks how likely it is that the coin remaining in that box is gold too. It is tempting to discard the Silver-Silver box and treat the other two boxes as equally likely, giving an answer of one-half. That reasoning overlooks the likelihood of the evidence. The Gold-Gold box can reveal gold in two distinct ways, while the Gold-Silver box can reveal gold in only one way. Conditioned on seeing gold, there are three equally likely gold-producing positions. Two positions come from the Gold-Gold box, so the exact probability of another gold coin is 2/3, approximately 66.67%. The calculator applies this exact theoretical probability to a chosen number of conditional trials. Multiplying the trial count by 2/3 gives the expected number of trials in which the unseen coin is gold. Because an expected value can be fractional while an observed count must be whole, the display rounds to the nearest whole outcome. It does not generate random samples, so repeated calculations remain deterministic. A real Monte Carlo simulation would fluctuate around this expectation and generally move closer to a two-thirds proportion as its trial count grows. The paradox teaches a broad lesson about conditional probability: possible explanations should be weighted by how readily each explanation produces the observation. This principle underlies Bayes' theorem and appears in diagnostic testing, screening, reliability analysis, and classification. The setup also requires clear sampling assumptions. Each box must have an equal initial chance of selection, and either coin in a selected box must have an equal chance of being drawn. Different selection weights would change the answer. Enter several trial counts to distinguish the fixed theoretical probability from the changing expected count, and remember that actual experimental totals can differ from expectation through random variation.