Three open wooden boxes holding conceptual gold and silver coins. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
The gold observation does not leave equally likely boxes
The classic setup has three equally likely boxes: gold–gold, gold–silver and silver–silver. Select one box uniformly, then one coin from that box uniformly. You observe gold. The probability that the other coin is gold is two-thirds, not one-half. The mixed box and the all-gold box both remain possible, but they were not equally likely to generate the observation.
Label the individual coins
Before observing color, the six individual coin draws are equally likely. Three of those draws reveal gold: the first gold coin from the all-gold box, the second gold coin from that box, and the gold coin from the mixed box. In two of these three observations, the other coin is gold. Conditioning removes the silver observations and then renormalizes the remaining ones.
| Box | Gold draw pathways | Other coin gold? |
|---|---|---|
| Gold–gold | 2 | Yes |
| Gold–silver | 1 | No |
| Silver–silver | 0 | Not eligible |
State what the observer actually did
“A gold coin was shown” is insufficient unless the showing mechanism is specified. If a host deliberately reveals gold whenever a chosen box contains gold, the mixed box always produces a gold reveal rather than producing one half the time. Under that different rule and equal box priors, the two surviving box types receive equal posterior weight. That is a different experiment, not a contradiction of the two-thirds result.
Check the observation model interactively
The Bertrand's Box Paradox tool uses a uniformly drawn coin within the selected box. It reports posterior weights for every box as well as the other-coin probability. You can also condition on observing silver. For unequal box-selection chances, read how biased box selection changes the answer. The general update is a small instance of Bayes' theorem; it is separate from Bertrand's random-chord paradox, which changes the geometry sampling mechanism.
Check the model behind your result
Use the Bertrand's Box Paradox for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26