What is the correct probability that the other coin in the box is gold after drawing a gold coin?

The probability is 2/3, not 1/2. Once a gold coin is revealed, the remaining possibilities are the gold-gold box (2/3 chance) and the gold-silver box (1/3 chance), since the silver-silver box is eliminated.

Why do many people intuitively think the probability is 1/2?

The intuitive 1/2 answer assumes the two remaining boxes are equally likely, but it ignores that the gold-gold box contains two gold coins, making it twice as likely to be the source of a gold draw compared to the mixed box.

Can I run a simulation to verify the 2/3 result?

The page includes a simulation mode that runs multiple trials to compare empirical results with the theoretical 2/3 probability, allowing users to see the law of large numbers in action.

Is the Bertrand's Box Paradox calculator on FreeSmartCalculator free to use?

Yes, the site states its tools are free to use without subscription requirements, making it accessible for students and casual users alike.

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Bertrand's Box Paradox

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About this tool

What Bertrand's Box Paradox does

Bertrand's Box Paradox is a classic probability puzzle involving three boxes: one with two gold coins, one with two silver coins, and one with one of each. The FreeSmartCalculator version presents an interactive calculator that walks users through the Bayes-based solution and explicitly highlights the common misconception between the 1/2 and 2/3 probability answers. The site emphasizes that its tools are free to use without subscription requirements, distinguishing it from paid or registration-gated alternatives. While the live page currently functions as a placeholder during migration to a unified calculator route, the stored notes confirm the tool's core purpose of demonstrating the paradox's conditional probability mechanics.

Step by step

How to use the FreeSmartCalculator Bertrand's Box Paradox

1
Select the box you have drawn from and the color of the coin revealed
2
The calculator displays the conditional probability that the other coin matches the revealed color
3
Review the step-by-step Bayes explanation provided on the page
4
Use the simulation mode to run multiple trials and compare empirical results with theoretical odds
5
Note that the interactive logic is temporarily in placeholder status during site migration

Is it right for you

Best for

This option suits learners and puzzle enthusiasts who want a clear, explanation-focused walkthrough of Bertrand's Box Paradox without needing to register or pay, especially those interested in understanding why the intuitive 1/2 answer is i

Limitations

The interactive calculator logic is currently a placeholder under site migration
Results and simulations may be limited or unavailable in the current page state
As a free site tool, advanced customization or extensive simulation runs may be restricted

Questions

Bertrand's Box Paradox FAQ

What is the correct probability that the other coin in the box is gold after drawing a gold coin?: The probability is 2/3, not 1/2. Once a gold coin is revealed, the remaining possibilities are the gold-gold box (2/3 chance) and the gold-silver box (1/3 chance), since the silver-silver box is eliminated.
Why do many people intuitively think the probability is 1/2?: The intuitive 1/2 answer assumes the two remaining boxes are equally likely, but it ignores that the gold-gold box contains two gold coins, making it twice as likely to be the source of a gold draw compared to the mixed box.
Can I run a simulation to verify the 2/3 result?: The page includes a simulation mode that runs multiple trials to compare empirical results with the theoretical 2/3 probability, allowing users to see the law of large numbers in action.
Is the Bertrand's Box Paradox calculator on FreeSmartCalculator free to use?: Yes, the site states its tools are free to use without subscription requirements, making it accessible for students and casual users alike.

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