Bertrand's Box Paradox Calculator
Explore the counterintuitive nature of conditional probability
Bertrand's Box Paradox is a famous probability puzzle that demonstrates how our intuition about probability can sometimes lead us astray. The paradox involves three boxes:
- Box 1: Contains two gold coins
- Box 2: Contains one gold coin and one silver coin
- Box 3: Contains two silver coins
The paradox arises when you randomly select a box, randomly draw one coin from it, and it turns out to be gold. Many people intuitively think the probability that you chose Box 1 (with two gold coins) is 2/3, but the correct answer is actually 1/3.
The problem follows these steps:
- You randomly select one of the three boxes
- You randomly draw one coin from the selected box
- You observe that the coin is gold
- You need to determine the probability that you selected Box 1 (the box with two gold coins)
The key to understanding this paradox is recognizing that this is a conditional probability problem and applying Bayes' Theorem correctly.
The most common misconception is thinking that since we drew a gold coin, it must be more likely that we chose Box 1. The reasoning usually goes:
- Box 1 has two gold coins, so it's more likely to be the source of our gold coin
- Box 2 has only one gold coin, so it's less likely to be the source
- Box 3 has no gold coins, so we can eliminate it
- Therefore, it must be twice as likely to be Box 1 than Box 2 (leading to the incorrect 2/3 probability)
This reasoning is flawed because it doesn't properly account for the conditional probabilities involved.
Let's solve this using Bayes' Theorem:
Prior Probabilities:
- P(Box 1) = 1/3
- P(Box 2) = 1/3
- P(Box 3) = 1/3
Likelihoods:
- P(Gold | Box 1) = 1 (both coins are gold)
- P(Gold | Box 2) = 1/2 (one of two coins is gold)
- P(Gold | Box 3) = 0 (no gold coins)
Using Bayes' Theorem:
P(Box 1 | Gold) = P(Gold | Box 1) × P(Box 1) / P(Gold)
Where P(Gold) = 1 × 1/3 + 1/2 × 1/3 + 0 × 1/3 = 1/2
Therefore: P(Box 1 | Gold) = (1 × 1/3) / (1/2) = 1/3
Understanding Bertrand's Box Paradox has important implications for:
- Medical Diagnosis: Understanding how test results affect the probability of having a condition
- Quality Control: Interpreting sampling results in manufacturing
- Scientific Research: Avoiding bias in experimental design and data interpretation
- Decision Making: Recognizing when intuition about probabilities might be misleading
- Risk Assessment: Properly evaluating conditional probabilities in risk scenarios