Why does the probability equal 2/3 instead of 1/2?

Because there are three gold coins total across the boxes, and two of them are in the double-gold box. Once a gold coin is revealed, the remaining unchosen coin is equally likely to be the other gold coin or the silver coin, making the double-gold box twice as likely as the mixed box.

Can I change the number of coins or boxes in the simulation?

The standard tool uses the fixed three-box configuration, but the Monte Carlo simulation lets you run many trials to see how often the theoretical 2/3 probability appears in practice.

Does the result change if I don't know which coin I drew?

No—the conditional probability remains 2/3 as long as you know at least one coin from the chosen box is gold, regardless of which specific coin it was.

How is this related to Bayes' theorem?

The paradox is a classic application of Bayes' rule: the prior probability of picking each box is 1/3, and the likelihood of drawing a gold coin updates that probability to 2/3 for the double-gold box.

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Bertrand's Box Paradox

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What is the chance of choosing the box containing only gold? Find the answer to Bertrand's box paradox with Omni!

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About this tool

What Bertrand's Box Paradox does

Bertrand's Box Paradox is a probability tool that calculates the conditional likelihood of selecting the box containing two gold coins given that a randomly drawn coin from that box is gold. The output presents the mathematically correct solution, which often contradicts common intuition, and includes a breakdown of the probability calculation process. Users input the setup parameters—three boxes with two gold, two silver, and one of each—to receive the theoretical probability instantly. The site's version distinguishes itself by offering a side-by-side comparison of theoretical odds and a Monte Carlo simulation, allowing users to run repeated trials and see empirical results converge on the theoretical answer. It also provides an explanatory article linking the paradox to Bayes' rule, turning a counterintuitive problem into a structured learning experience about conditional probability and perception of odds.

Step by step

How to use the Omni Calculator Bertrand's Box Paradox

1
Select the box you initially choose from the three available options
2
Draw a coin at random from the selected box and observe its type
3
The calculator displays the conditional probability that the other coin in the box is also gold
4
Click 'Let's recreate this situation' to run a new simulation or 'Share result' to save your outcome

Is it right for you

Best for

Students and curious learners who want to resolve a famous probability puzzle and understand why the intuitive answer is usually wrong.

Limitations

Results rely on the classic three-box setup and may not apply to variations with different numbers of boxes or coins
The Monte Carlo simulation provides estimates rather than exact theoretical proofs
The tool focuses on a single paradox scenario and does not offer broader probability distributions

Questions

Bertrand's Box Paradox FAQ

Why does the probability equal 2/3 instead of 1/2?: Because there are three gold coins total across the boxes, and two of them are in the double-gold box. Once a gold coin is revealed, the remaining unchosen coin is equally likely to be the other gold coin or the silver coin, making the double-gold box twice as likely as the mixed box.
Can I change the number of coins or boxes in the simulation?: The standard tool uses the fixed three-box configuration, but the Monte Carlo simulation lets you run many trials to see how often the theoretical 2/3 probability appears in practice.
Does the result change if I don't know which coin I drew?: No—the conditional probability remains 2/3 as long as you know at least one coin from the chosen box is gold, regardless of which specific coin it was.
How is this related to Bayes' theorem?: The paradox is a classic application of Bayes' rule: the prior probability of picking each box is 1/3, and the likelihood of drawing a gold coin updates that probability to 2/3 for the double-gold box.

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