Bertrand's Box Paradox Calculator
Explore the famous probability paradox and understand why the answer is 2/3, not 1/2
Explore Bertrand's Box Paradox
The Paradox
You have three boxes:
- • Box 1: Two gold coins (🟡🟡)
- • Box 2: Two silver coins (⚪⚪)
- • Box 3: One gold, one silver (🟡⚪)
If you randomly pick a box and draw a gold coin, what's the probability that the other coin in the box is also gold?
Interactive Demonstration
Theoretical Solution Using Bayes' Theorem
P(GG|G) = P(G|GG) × P(GG) / P(G)
Given:
- • P(G|GG) = 1 (drawing gold from GG box)
- • P(GG) = 1/3 (choosing GG box)
- • P(G) = 1/2 (drawing gold overall)
Calculation:
P(G) = 1×(1/3) + (1/2)×(1/3) + 0×(1/3)
P(G) = 1/3 + 1/6 = 1/2
P(GG|G) = (1 × 1/3) / (1/2) = 2/3
Final Answer: 66.7% or 2/3
Why Not 1/2?
Common misconception: "There are 2 boxes with gold, so 50% chance."
The key insight: Each gold coin has different probabilities of being drawn first.
The GG box has 2 ways to draw gold first, while the GS box has only 1 way!
Quick Facts
Created by Joseph Bertrand in 1889
Similar to the Monty Hall problem
Demonstrates conditional probability
Answer is always 2/3, never 1/2
Understanding Bertrand's Box Paradox
The Intuitive (Wrong) Answer
"Since I drew gold, I know I'm not in the SS box. That leaves GG and GS boxes. So there's a 50% chance the other coin is gold."
This reasoning seems logical but fails because it doesn't account for the different ways each box can produce a gold coin on the first draw.
The Key Insight
Think of it this way: there are 6 coins total (2 in each box). When you draw gold, you could have drawn any of the 3 gold coins. Two of these belong to the GG box, and only one belongs to the GS box.
Step-by-Step Solution
Step 1: List all possible first draws that result in gold:
- • GG box, first coin: Gold
- • GG box, second coin: Gold
- • GS box, gold coin: Gold
Step 2: Count favorable outcomes (other coin is gold):
- • GG box, first coin → second coin is Gold ✓
- • GG box, second coin → first coin is Gold ✓
- • GS box, gold coin → other coin is Silver ✗
Step 3: Calculate probability:
2 favorable / 3 total = 2/3 ≈ 66.67%
Why This Matters
Bertrand's Box Paradox illustrates how our intuition about probability can be misleading. It's a classic example of how conditional probability works and why we need mathematical tools like Bayes' theorem to get the correct answer.
This type of reasoning appears in many real-world scenarios, from medical testing to machine learning algorithms, making it crucial to understand the underlying principles.