# Bertrand’s Box Paradox Calculator

URL: https://speedycalculator.com/calculators/statistics/bertrand-box-paradox/
Last updated: 2026-10-30

> The Bertrand's box paradox calculator finds the chance that the second coin in a randomly chosen box matches the first coin you pulled out, for any mix of gold-gold, silver-silver and gold-silver boxes, and checks the answer with thousands of simulated draws.

## Inputs

- The first coin you drew was
- Boxes with two gold coins
- Boxes with two silver coins
- Boxes with one gold and one silver coin
- Simulated draws

## Results

- Chance the other coin has the same colour
- As a fraction
- Chance it is the other colour
- Simulated frequency
- Simulated draws that showed your colour first
- One random draw
- Exact answer, simulation and the naive guess

## Formula

```
P(other coin gold | first coin gold) = 2·n_GG ÷ (2·n_GG + n_GS) · with one box of each kind: 2/3
```

## Explanation

Joseph Bertrand described the puzzle in 1889. There are three boxes: one holds two gold coins, one holds two silver coins and one holds a gold and a silver coin. You choose a box at random, take one coin out without looking inside and it turns out to be gold. What is the chance that the other coin in the same box is gold too?

Most people answer one half: the box is either the gold-gold one or the mixed one, so it seems like a coin toss. The right answer is two thirds. The two boxes are not equally likely once you have seen a gold coin, because the gold-gold box offers two gold coins you could have drawn while the mixed box offers only one. Of the three gold coins you might be holding, two have a gold partner.

This calculator lets you change the number of boxes of each kind and the colour you saw, gives the exact conditional probability as a percentage and a fraction, and runs a simulation so you can watch the frequency settle near the exact value. Press the button to open one more box and follow a single draw.

### How to use the Bertrand's Box Paradox Calculator

- **The first coin you drew was**
- **Boxes with two gold coins**
- **Boxes with two silver coins**
- **Boxes with one gold and one silver coin**
- **Simulated draws**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The exact answer counts coins rather than boxes: every coin is equally likely to be the one drawn, so P = (gold coins whose partner is gold) ÷ (all gold coins). The simulation picks a random box and a random coin from it many times, keeps only the draws that showed your colour first and reports how often the second coin matched.

### Worked example

Label the three gold coins: G1 and G2 in the gold-gold box, G3 in the mixed box. Each is equally likely to be the coin in your hand. If you hold G1 or G2 the other coin is gold; if you hold G3 it is silver. Two favourable cases out of three give 2/3 ≈ 66.67%. With 2 gold-gold boxes, 1 silver-silver box and 3 mixed boxes the calculator gives 2·2 ÷ (2·2 + 3) = 4/7 ≈ 57.14%.

### Why counting boxes fails

Before you look, each box has a 1/3 chance. A gold coin can come from the gold-gold box with certainty but from the mixed box only half the time, so the evidence is twice as likely under the gold-gold box. Multiplying the priors by these likelihoods gives weights 1/3 : 1/6 = 2 : 1, which is the 2/3 answer.

## Worked example

Inputs — The first coin you drew was: Gold; Boxes with two gold coins: 1; Boxes with two silver coins: 1; Boxes with one gold and one silver coin: 1; Simulated draws: 10,000.

Results — **Chance the other coin has the same colour = 66.67 %**; As a fraction = 2/3; Chance it is the other colour = Chance the other coin is silver: 33.33%; Simulated frequency = 67.17 %; Simulated draws that showed your colour first = 5,001; One random draw = You opened box 2 (two silver coins) and pulled out a silver coin. The other coin in that box is silver. This draw did not show gold first, so it does not count towards the question – draw again..

## Key terms

- **Conditional probability**: The probability of an event given that another event has happened, written P(A | B).
- **Bayes' theorem**: P(A | B) = P(B | A) × P(A) / P(B), the rule for updating a probability after seeing evidence.
- **Prior probability**: The chance before the evidence – here 1/3 for each box.
- **Posterior probability**: The chance after the evidence – 2/3 for the gold-gold box once a gold coin is seen.

## FAQ

**What is the answer to Bertrand's box paradox?**
2/3, about 66.67%. Seeing a gold coin makes the gold-gold box twice as likely as the mixed box, because it has two gold coins you could have drawn.

**Why is the intuitive answer of 1/2 wrong?**
It treats the two remaining boxes as equally likely. They were equally likely before the draw, but the gold coin is evidence that favours the box with more gold coins.

**How does Bayes' theorem solve it?**
P(GG | gold) = P(gold | GG) × P(GG) / P(gold) = 1 × (1/3) / (1/2) = 2/3.

**What changes if there are two mixed boxes?**
With one gold-gold box and two mixed boxes there are four gold coins, two of them in the gold-gold box, so the answer drops to 2/4 = 50%.

**Is this the same as the Monty Hall problem?**
Not the same game, but the same lesson: new information changes the odds unevenly, and counting equally likely outcomes beats counting boxes or doors.

**How many simulated draws are enough?**
With 10,000 draws about half count (those showing gold), so the simulated value is usually within about 1.3 percentage points of 66.67%.


## Sources

- [Wikipedia – Bertrand's box paradox](https://en.wikipedia.org/wiki/Bertrand%27s_box_paradox)
- [Penn State STAT 414 – Introduction to Probability Theory](https://online.stat.psu.edu/stat414/)
- [Khan Academy – Statistics and probability](https://www.khanacademy.org/math/statistics-probability)