How this calculator works
Condition on the individual coins that could have produced the observation. A same-colour box contributes two matching coins, while a mixed box contributes only one. Equal prior probabilities for boxes therefore do not remain equal after a colour is seen. The result depends on uniform box selection and a uniformly chosen coin within it.
Formula
P(same colour | observed A) = 2 × AA / (2 × AA + AB). P(observe A) = (2 × AA + AB) / [2 × (AA + BB + AB)]. Exchange AA and BB for colour B.
Worked example
With one AA box, one BB box and one mixed box, an A observation could be either coin in AA or the A coin in AB. Two of those three possibilities leave another A coin, so the answer is 2/3.
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Frequently asked questions
Why is the answer not one half?
The observation is twice as likely from a two-A box as from a mixed box. Counting only the two remaining box types misses this weighting.
Does revealing a chosen compartment change the answer?
It can. A deliberate reveal rule can assign different probabilities to the possible observations. This tool assumes a randomly selected coin.
Can I use different numbers of boxes?
Yes. The counts represent equal-weight boxes of each type. They must be non-negative whole numbers, and the observed colour must exist.