Bertrand's Box Paradox Calculator

Understanding the Setup

Bertrand's paradox begins with three indistinguishable boxes containing coins of two types:

• Box A: Two gold coins (GG)
• Box B: Two silver coins (SS)
• Box C: One gold and one silver coin (GS)

You randomly select one box without seeing its contents. From that box, you blindly draw a single coin. The coin you drew is gold. Now the question becomes: what is the probability that the remaining coin in your chosen box is also gold?

The paradox arises because our intuition suggests the answer should be 1/2—after all, the remaining coin must be either gold or silver, giving it seemingly equal odds. However, the actual probability is 2/3. This discrepancy reveals a fundamental misunderstanding about how drawing evidence constrains probability.

The Mathematical Solution

Using Bayes' theorem, we can derive the exact probability. We need to consider how many ways we can draw a gold coin from each box, then determine which scenarios allow a second gold coin.

• P(GG | Gold drawn) — Posterior probability that the box contains two gold coins, given that you drew gold
• P(Gold | GG) — Likelihood of drawing gold from the two-gold box, which equals 1 (certainty)
• P(GG) — Prior probability of selecting the two-gold box, which is 1/3
• P(Gold) — Marginal probability of drawing any gold coin, which equals 1/2

Why Intuition Fails: The Enumeration Argument

The clearest way to see why 2/3 is correct is to enumerate all possible gold-coin draws:

• Scenario 1: You pick Box A (GG) and draw the first gold coin
• Scenario 2: You pick Box A (GG) and draw the second gold coin
• Scenario 3: You pick Box C (GS) and draw the single gold coin

Each scenario is equally likely when you condition on drawing gold. In two of these three scenarios (1 and 2), the remaining coin is gold. In only one scenario (3), the remaining coin is silver.

This reveals the trap: when you drew gold, you made it much more likely that you selected Box A, because Box A offers two chances to draw gold while Box C offers only one. The evidence of drawing gold thus shifts the probability distribution over the boxes themselves.

Connection to Conditional Probability

Bertrand's paradox is fundamentally a lesson in conditional probability. Many people incorrectly apply the principle of indifference—assuming that because two outcomes remain possible, they must be equally probable. This fails when those outcomes have different paths to the observed evidence.

Consider the symmetry argument: if you had instead drawn a silver coin, by identical logic the probability that the remaining coin is silver would be 2/3. This demonstrates that the result is not an artifact of choosing gold, but rather reflects how the sampling process itself creates asymmetry.

Bayes' rule formalises this insight: it updates your belief about which box you selected based on the evidence of the coin you drew, revealing that you are twice as likely to have chosen the two-gold box than the mixed box.

Common Pitfalls and Insights

Understanding why people typically misjudge this problem helps sharpen your conditional probability intuition.

1. Assuming equal likelihood of remaining outcomes — The remaining coin is not equally likely to be gold or silver. The gold coin you drew provides strong evidence about which box you selected, heavily favouring the all-gold box over the mixed box. Always ask: does the evidence I observed make some scenarios more probable than others?
2. Confusing the sample space — Some calculate odds as 1/2 by imagining only two possible boxes remain after drawing gold—either the all-gold or mixed box. But you must account for the fact that there are two indistinguishable gold coins in the all-gold box, each independently capable of being drawn. This creates an asymmetry in likelihood.
3. Forgetting about prior probabilities — The probability of selecting each box initially is 1/3. When you draw gold, this prior combines with the likelihood of drawing gold from each box type to produce a posterior probability. Ignoring the prior leads to the 1/2 error.
4. Not simulating enough trials — Running a simulation with insufficient repetitions may show results near 1/2 due to random variation. Use at least 10,000 trials to see the true 2/3 probability converge, especially if testing this empirically.