A hand selecting among weighted wooden coin boxes with a balance nearby. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Relative weights describe the box-selection stage
Equal box priors are an assumption of the classic puzzle, not a physical law. Suppose a mechanism selects the gold–gold box with relative weight 1, the gold–silver box with weight 4 and the silver–silver box with weight 1. The normalized prior chances are 1/6, 4/6 and 1/6. Within whichever box is selected, one coin is still drawn uniformly.
Multiply each prior by its observation chance
For an observed gold coin, the unnormalized posterior weights are 1 × 1 for gold–gold, 4 × 1/2 for gold–silver and 1 × 0 for silver–silver. Their total is 3. Gold–gold therefore receives posterior weight 1/3, while gold–silver receives 2/3. The probability that the other coin is gold is now one-third. More mixed boxes in the selection process make the gold observation less persuasive evidence for an all-gold box.
Scaling all weights changes nothing
Entering 10, 40 and 10 gives the same posterior as entering 1, 4 and 1. Only relative weights matter. A zero weight removes a box from selection, but all three weights cannot be zero. If you remove every box containing the observed color, that observation becomes impossible under the model and the conditional answer is undefined.
Keep selection bias separate from reveal bias
Changing the chance of selecting a box is different from letting a host choose which coin to reveal. The box-paradox calculator varies the former while keeping the within-box draw uniform. It does not silently change the host protocol. Write both stages when comparing answers from different puzzle descriptions. The coin-observation guide explains why equal surviving boxes are not the same as equal observation pathways. For another example in which the information protocol changes the conditional probability, compare the Boy or Girl Paradox tool.
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