Bertrand's Box Paradox Calculator
Explore the counterintuitive probability puzzle involving three boxes of coins. Adjust how many boxes contain two gold coins, one gold and one silver coin, or two silver coins, then observe how the posterior probability changes once you reveal a gold coin.
Probability Other Coin is Gold
66.67%
P(Golden pair | observed gold)
Selected Box was GG
66.67%
Posterior probability
Selected Box was GS
33.33%
Posterior probability
| Box Type | Prior Probability | Probability of Drawing Gold | Contribution to Evidence |
|---|---|---|---|
| Two Gold Coins (GG) | 33.33% | 100% | 33.33% |
| One Gold & One Silver (GS) | 33.33% | 50% | 16.67% |
| Two Silver Coins (SS) | 33.33% | 0% | 0% |
How to Use This Calculator
- Enter how many boxes contain two gold coins, how many contain one of each, and how many contain two silver coins.
- Imagine you choose a box uniformly at random and draw one coin at random. The calculator conditions on the event that the drawn coin is gold.
- Read the resulting posterior probability that the hidden coin is also gold. The paradox shows that it is typically higher than intuitive guesses.
- Experiment with different box counts to see how the relative frequencies change the conditional probability.
Formula
- P(GG) — Probability of picking a box with two gold coins on the first draw.
- P(GS) — Probability of picking a box with one gold and one silver coin.
- 1 — If the box is GG, every coin drawn is gold.
- 0.5 — If the box is GS, the chance of drawing the gold coin on the first try.
- The SS box contributes 0 because no gold coin can be drawn from it.
Full Description
Bertrand's Box Paradox challenges our intuition about conditional probability. Even though there may be only one box with two gold coins and one box with one gold and one silver coin, once you observe a gold coin the likelihood that you selected the all-gold box is two times higher than selecting the mixed box. The event of seeing gold provides evidence that favors boxes containing more gold coins.
This calculator applies Bayes' theorem to weigh each scenario. It multiplies the prior chance of choosing each box by the probability of observing gold from that box, then renormalizes to obtain the posterior. Altering the number of GG boxes or GS boxes changes the posterior dramatically and reveals how evidence updates beliefs.
Frequently Asked Questions
Why isn't the answer 50/50?
Drawing a gold coin makes it more likely you picked a box containing more gold coins. The all-gold box guarantees a gold draw, whereas the mixed box only offers a gold coin half the time. This asymmetry tilts the probability toward the all-gold box.
What happens if there are no gold coins?
If none of the boxes contain gold, the event of observing a gold coin has zero probability, so the conditional probability is undefined. The calculator highlights this situation to show that the paradox requires at least one gold coin to make sense.
Can I add more than three boxes?
Yes. The calculation only depends on how many boxes fall into each category. You can extend the paradox by increasing the counts; the formulas scale automatically.
How does this relate to Bayes' theorem?
Bayes' theorem updates the probability of each box type after observing a gold coin. The posterior probability of the GG box is proportional to its prior probability times the likelihood of seeing gold from that box.