What is the probability that the second ball is gold given the first was gold in the classic Bertrand's Box Paradox?

The probability is 2/3. Even though it seems like a 50/50 chance after seeing one gold ball, the original box with two gold bars is twice as likely to be the one selected.

Can I create my own box configurations beyond the preset scenarios?

Yes, the tool allows custom scenarios where you specify the number of gold and silver balls in each of the three boxes, up to the entry limits shown on the page.

How many Monte Carlo simulations should I run for reliable results?

The default is 100,000 simulations for the classic scenario; larger numbers reduce empirical variance, but even 10,000 trials typically show convergence near the theoretical 2/3 probability.

Does the tool show the step-by-step reasoning for the conditional probability?

No, it provides the final calculated probability and simulation frequency; the focus is on the result and empirical comparison rather than a walkthrough of the Bayes' theorem steps.

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Bertrand's Box Paradox

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ToolDone offers a dedicated Bertrand's Box Paradox calculator that computes conditional probability for the classic b...

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About this tool

What Bertrand's Box Paradox does

Bertrand's Box Paradox is a calculator for conditional probability problems drawn from the classic three-box setup. Users select a scenario — the original paradox, a symmetric configuration, or a custom arrangement — and the tool computes the probability that the second ball is gold given the first drawn was gold. A Monte Carlo simulation runs alongside the theoretical result, showing empirical frequencies that converge on the expected value. The interface displays current simulation counts and lets users reset or adjust parameters without leaving the page. While the primary focus is the gold-silver ball draw, the site also hosts a suite of standard deviation and mode calculators, positioning the tool within a broader statistics hub. Compared with other free calculators, ToolDone’s version distinguishes itself by offering multiple selectable scenarios and a running simulation counter, allowing side-by-side comparison of theoretical and empirical probabilities in one place.

Step by step

How to use the ToolDone Bertrand's Box Paradox

1
Select a paradox scenario from the dropdown or build a custom box configuration
2
Choose the number of Monte Carlo simulations or accept the default count
3
Click Calculate Probability to view the conditional probability result
4
Review the simulation frequency display to compare empirical results with the theoretical answer
5
Use Reset Calculator to clear inputs and start a new configuration

Is it right for you

Best for

Students, instructors, or anyone exploring conditional probability who wants to see both the theoretical solution and a visual simulation side by side.

Limitations

No unit switching between percentages and fractions
Simulation counts reset manually; long runs may be needed for tight convergence
Custom scenarios are limited to the ball-count entries provided

Questions

Bertrand's Box Paradox FAQ

What is the probability that the second ball is gold given the first was gold in the classic Bertrand's Box Paradox?: The probability is 2/3. Even though it seems like a 50/50 chance after seeing one gold ball, the original box with two gold bars is twice as likely to be the one selected.
Can I create my own box configurations beyond the preset scenarios?: Yes, the tool allows custom scenarios where you specify the number of gold and silver balls in each of the three boxes, up to the entry limits shown on the page.
How many Monte Carlo simulations should I run for reliable results?: The default is 100,000 simulations for the classic scenario; larger numbers reduce empirical variance, but even 10,000 trials typically show convergence near the theoretical 2/3 probability.
Does the tool show the step-by-step reasoning for the conditional probability?: No, it provides the final calculated probability and simulation frequency; the focus is on the result and empirical comparison rather than a walkthrough of the Bayes' theorem steps.

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