Bertrand Box Paradox
Explore Bertrand's box paradox — simulate three boxes with gold and silver coins. Discover why P(second gold) = 2/3, not 1/2, with interactive Monte Carlo trials.
About This Calculator
Bertrand's box paradox is a classic probability puzzle that exposes the gap between intuition and mathematical reasoning in conditional probability. Named after the French mathematician Joseph Bertrand who published it in 1889, the paradox asks: given 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 random box and draw a coin at random. If that coin is gold, what is the probability that the other coin in the same box is also gold? Most people instinctively answer 1/2, but the correct answer is 2/3.
The reasoning becomes clear when you count the gold coins rather than the boxes. There are three gold coins total: two belong to the GG box and one belongs to the GS box. Each gold coin is equally likely to be the one you drew. In two of those three equally-likely scenarios, the other coin in the box is also gold. This calculator runs a Monte Carlo simulation that randomly selects boxes and draws coins across thousands of trials, allowing you to watch the empirical probability converge to 2/3. A full Bayes theorem breakdown is shown alongside the results for deeper understanding.
The paradox is closely related to other famous probability puzzles such as the Monty Hall problem and the Boy or Girl paradox, all of which demonstrate Bayesian updating — the process of revising probabilities as new information becomes available. Whether you are a student learning probability, a teacher looking for a classroom demonstration, or simply curious about counterintuitive mathematics, this interactive tool brings Bertrand's classic puzzle to life.
Frequently Asked Questions
What is Bertrand's box paradox?
Bertrand's box paradox is a probability problem that demonstrates how our intuition about conditional probability can be misleading. There are three boxes: one with two gold coins, one with two silver coins, and one with one gold and one silver coin. You pick a random box and draw a coin at random. If it is gold, the probability that the other coin in the box is also gold is 2/3, not 1/2 as common intuition suggests.
Why is the answer 2/3 and not 1/2?
The answer is 2/3 because when you draw a gold coin, you are more likely to have picked the box with two gold coins. There are three gold coins in total: two from the GG box and one from the GS box. Since each gold coin is equally likely to be drawn, two of the three possible gold coin draws come from the GG box, giving a 2/3 probability that the other coin in the box is also gold.
How does Bayes theorem solve Bertrand's box paradox?
Bayes theorem calculates P(GG|G) = P(G|GG) × P(GG) / P(G). P(G|GG) = 1 (the GG box always yields gold), P(GG) = 1/3 (one of three boxes), and P(G) = 1/2 (overall probability of drawing gold across all boxes). Plugging these in: (1 × 1/3) / (1/2) = 2/3, confirming the counterintuitive result.
What is the difference between Bertrand's box paradox and the Monty Hall problem?
Both are counterintuitive probability puzzles involving conditional probability. In Bertrand's box paradox, you select a box and draw a coin; after seeing a gold coin, you compute the probability the other coin is gold. In the Monty Hall problem, you pick a door, the host opens a losing door, and you decide whether to switch. Both illustrate Bayesian updating but in different game structures.
Can Bertrand's box paradox be verified with a simulation?
Yes. By running thousands of random trials — randomly selecting a box, drawing a coin, and checking whether the second coin is gold when the first is gold — the empirical probability converges to 2/3. Our calculator runs such a simulation, and you can adjust the number of trials to see the convergence.
Who invented Bertrand's box paradox?
The paradox was introduced by Joseph Bertrand, a French mathematician, in his 1889 book 'Calcul des probabilités'. He used the problem to illustrate how intuitive reasoning about probability can lead to incorrect conclusions.
What is the probability of drawing a gold coin from the mixed box?
The probability of drawing a gold coin from the mixed box (GS) is 1/2, since one of the two coins is gold. If you draw a gold coin from the GS box, the other coin is silver. This contrasts with the GG box, where a gold draw guarantees the other coin is also gold.
How is the overall probability of drawing gold calculated in Bertrand's box paradox?
The overall probability P(G) of drawing gold is the weighted sum across all three boxes: from the GG box (1 × 1/3 = 1/3), from the GS box (1/2 × 1/3 = 1/6), and from the SS box (0 × 1/3 = 0). Summing gives P(G) = 1/3 + 1/6 = 1/2.