Bertrand's Box Paradox Calculator

Bertrand's Box Paradox is a classic probability puzzle introduced by Joseph Bertrand in 1889. There are three boxes: one with two gold coins (GG), one with two silver coins (SS), and one with one gold and one silver coin (GS). You pick a box at random and draw one coin — it's gold. The paradox asks: what is the probability that the other coin in the same box is also gold? Most people intuitively say 1/2, but the correct answer is 2/3.

The intuition of 1/2 comes from thinking there are only two possible boxes left (GG or GS), each equally likely. However, you must account for how you arrived at drawing a gold coin. There are three gold coins total, and two of them are in the GG box. So given that you drew a gold coin, the chance you're in the GG box is 2 out of 3 — not 1 out of 2.

Using Bayes' rule: P(GG | gold drawn) = P(gold drawn | GG) × P(GG) / P(gold drawn). P(gold | GG) = 1, P(GG) = 1/3, and P(gold drawn) = 1/2 (since 3 of 6 coins total are gold). This gives (1 × 1/3) / (1/2) = 2/3. Bayes' rule formalises the correct conditional probability by weighting each scenario by its likelihood of producing the observed outcome.

The correct probability that the other coin in the box is gold — given you already drew a gold coin — is 2/3 (approximately 66.67%). This can be verified by reasoning, Bayes' theorem, or through repeated simulation, which this calculator demonstrates.

In a simulation, you randomly pick a box and randomly pick a coin from it. You record only the trials where the drawn coin matches your condition (e.g., gold). Among those valid trials, you count how many came from the same-type box (GG). As the number of trials grows, this ratio converges to 2/3, confirming the theoretical answer.

Yes — both are conditional probability puzzles where human intuition systematically gives the wrong answer. In both cases, failing to properly account for how the observed evidence was generated leads people to incorrectly assume equal probability among remaining options. The underlying mathematics in both is Bayes' rule.

By symmetry, yes — if you draw a silver coin first, the probability that the other coin in that box is also silver is also 2/3. The same reasoning applies: there are three silver coins total, two of which are in the SS box, so given a silver coin was drawn, the probability you're in the SS box is 2/3.

This is the Law of Large Numbers. With few trials, random variation can cause the simulated probability to stray far from 2/3. As you increase the number of trials (try 100,000 or more), the simulated result converges reliably toward the theoretical value of 66.67%, reducing the impact of random noise.