Bertrand's Box Paradox

How Bertrand's Box Paradox Works?

Bertrand's Box Paradox uses conditional probability to determine which type of box was selected after one coin is observed. Let $a$ be the number of double-gold boxes, $b$ the number of double-silver boxes, and $c$ the number of mixed-coin boxes. A box is selected at random, and then one of its two coins is selected at random.

The likelihood of observing gold is 1 for a double-gold box, 0 for a double-silver box, and $\frac{1}{2}$ for a mixed box. The corresponding likelihoods for observing silver are 0, 1, and $\frac{1}{2}$. Bayes' theorem updates the initial box probabilities using these likelihoods.

For an observed gold coin, the posterior probabilities are calculated from the relative weights $a$, 0, and $\frac{c}{2}$.

For an observed silver coin, the calculation is symmetric: the double-gold probability is zero, while the double-silver and mixed probabilities are based on $b$ and $\frac{c}{2}$.

The remaining-coin probabilities follow from the box type. After observing gold, a double-gold box leaves a gold coin and a mixed box leaves silver. After observing silver, a double-silver box leaves silver and a mixed box leaves gold. The calculator reports all five posterior results as percentages, and the three box-type probabilities for each observation sum to 100%.

Practical Use Cases

Students of probability: Students can enter the classic one-box-of-each configuration and observe why seeing a gold coin does not make the double-gold and mixed boxes equally likely. With one box of each type and gold observed, the calculator returns a 66.67% probability for double-gold and 33.33% for mixed.

Teachers and instructors: Educators can vary the numbers of double-gold, double-silver, and mixed boxes to demonstrate how prior proportions affect posterior probabilities. For example, increasing the number of mixed boxes shows how repeated opportunities to observe either color change the result.

Probability enthusiasts: Puzzle solvers can test both gold and silver observations, compare the symmetrical formulas, and verify that the posterior box probabilities add to 100%. The remaining-coin outputs also make the selection bias visible.

Statistics and data-science learners: Learners can use the calculator as a small discrete Bayesian model, treating box type as the hidden hypothesis and the observed coin color as evidence. Changing the input counts illustrates how priors and likelihoods combine.

Game and experiment designers: Designers can model urn, box, or concealed-object experiments with unequal numbers of each box type. The calculator helps predict the information gained from revealing one randomly selected coin before designing a classroom demonstration or puzzle.

Accuracy and Limitations

The calculator gives mathematically exact conditional-probability results for integer counts from 1 through 1,000 for each box category, subject to normal display rounding in the reported percentages. It assumes every box is selected with equal probability, each box contains exactly two coins of the stated composition, and either coin in a selected box is equally likely to be drawn. It does not model biased box selection, unequal coin-drawing probabilities, additional observations, or boxes containing other compositions; the three box counts must also be positive integers within the allowed input range.

Quick Examples