Bertrand Box Paradox Calculator

What is Bertrand's Box Paradox?

Bertrand's Box Paradox is a classic puzzle in conditional probability first proposed by Joseph Bertrand in 1889. It demonstrates how intuitive reasoning often fails when calculating conditional probabilities.

The Classic Setup

Imagine three identical boxes:

• Box 1: Contains 2 gold coins (GG).
• Box 2: Contains 2 silver coins (SS).
• Box 3: Contains 1 gold coin and 1 silver coin (GS).

You choose a box at random and draw one coin at random without looking inside. If the drawn coin is gold, what is the probability that the other coin in the box is also gold?

Mathematical Formula & Explanation

Many people intuitively guess 50% ($1/2$), assuming that having drawn a gold coin leaves only Box 1 (GG) and Box 3 (GS) as possibilities. However, the correct answer is 66.67% ($2/3$).

Using Bayes' Theorem:

$$P(\text{GG} \mid \text{Gold}) = \frac{P(\text{Gold} \mid \text{GG}) \cdot P(\text{GG})}{P(\text{Gold})}$$

Where:

• $P(\text{GG}) = \frac{1}{3}$
• $P(\text{Gold} \mid \text{GG}) = 1$
• $P(\text{Gold}) = \frac{1}{3} \cdot 1 + \frac{1}{3} \cdot 0 + \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{2}$

Thus:

$$P(\text{GG} \mid \text{Gold}) = \frac{1 \cdot (1/3)}{1/2} = \frac{2}{3}$$

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