Why is P(other gold) = 2/3 and not 1/2?

When you draw gold, you've eliminated the SS box. You're equally likely to have picked from GG or GS. But in GG there are 2 gold coins — so 2 of 3 'gold draws' come from GG. Hence P(other gold) = 2/3.

How does this relate to Monty Hall?

Both involve conditional probability and counterintuitive results. Monty Hall: switching wins 2/3. Bertrand's box: P(other gold) = 2/3. Both defy the naive 1/2 intuition.

What are the three boxes?

Box 1: two gold coins (GG). Box 2: two silver coins (SS). Box 3: one gold, one silver (GS). You pick a box at random, draw one coin, and it's gold.

Why run a simulation?

Monte Carlo simulation verifies the theoretical 2/3. With many trials, the empirical probability converges to the theoretical value. Great for building intuition.

What is the Bayesian proof?

P(GG|gold) = P(gold|GG)×P(GG) / P(gold) = (1×1/3) / (1/2) = 2/3. P(gold) = 1/3 + 0 + 1/6 = 1/2 (from GG, SS, GS).

Joseph Bertrand (1822–1900), French mathematician. His 1889 book Calcul des probabilités introduced several probability paradoxes, including this box paradox and the chord paradox.

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PROBABILITYProbability TheoryStatistics Calculator

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Bertrand's Box — Why P(other gold) = 2/3

Three boxes: GG, SS, GS. You draw gold. The other coin is gold 2/3 of the time — not 1/2. Count the coins: 2 of 3 gold draws come from GG.

Run SimulationMonte Carlo verification

Why This Statistical Analysis Matters

Why: Classic probability paradox. Intuition says 1/2; math says 2/3. Same reasoning as Monty Hall — conditional probability defies intuition.

How: Run Monte Carlo simulation. Or follow the Bayesian proof: P(GG|gold) = P(gold|GG)×P(GG)/P(gold) = (1×1/3)/(1/2) = 2/3.

●2 of 3 gold draws from GG
●P(gold) = 1/2
●Bayes: 2/3

Sources:Bertrand's Calcul des probabilités (1889)Wikipedia - Bertrand's box paradox

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PROBABILITY PARADOXThree boxes, gold/silver

Bertrand's Box — Why P(other gold) = 2/3

Three boxes: GG, SS, GS. You draw gold. The other coin is gold 2/3 of the time — not 1/2. Count the coins: 2 of 3 gold draws come from the GG box.

Bertrand Chord →Boy or Girl →

Examples — Click to Load

The 3 Boxes

Box 1 (GG)

🪙 🪙

Both gold — other is gold ✓

Box 2 (SS)

⬜ ⬜

Both silver — can't draw gold

Box 3 (GS)

🪙 ⬜

1 gold, 1 silver — other is silver

Simulation

Trials

Custom (1–1,000,000)

bertrand_box.sh

THEORETICAL 2/3

$ P(other is gold | drew gold)

2/3 = 66.67%

Common wrong answer: 1/2 — ignores that GG box is twice as likely to produce gold.

Bertrand's Box Paradox

P(other gold | drew gold)

2/3

Theoretical: 66.67%

numbervibe.com/calculators/statistics/bertrand-box-paradox

Calculation Breakdown

SETUP

Three boxes

GG, SS, GS

ext{Box} 1: 2 ext{gold}, ext{Box} 2: 2 ext{silver}, ext{Box} 3: 1 ext{gold} + 1 ext{silver}

COMPUTATION

P(draw gold)

1/2

3 ext{gold} ext{coins} / 6 ext{total} = 1/2

COMPUTATION

Gold coins: 2 in GG, 1 in GS

2 of 3 from GG

ext{When} ext{you} ext{draw} ext{gold}, 2/3 ext{of} ext{the} ext{time} ext{you} ext{picked} ext{GG} ext{box}

RESULT

P(other gold | drew gold)

2/3

P( ext{GG}| ext{gold}) = P( ext{gold}| ext{GG}) imes P( ext{GG})/P( ext{gold}) = 1 imes rac{1/3}{1/2} = 2/3

NOTE

Common wrong answer

1/2

ext{Ignores} ext{that} ext{GG} ext{box} ext{is} ext{twice} ext{as} ext{likely} ext{to} ext{produce} ext{gold} ext{draw}

Outcome Distribution

P(Box | drew gold)

⚠️For educational and informational purposes only. Verify with a qualified professional.

📈 Statistical Insights

2/3

P(other gold | drew gold)

— Bertrand 1889

2 gold draws from GG box

— Count coins

Bayes

P(GG|gold)=2/3

— Proof

Key Takeaways

• The answer is 2/3, not 1/2 — the all-gold box is twice as likely to produce a gold draw
• Equivalent to Monty Hall and Boy or Girl paradox — conditional probability depends on the observation process
• P(GG | gold) = P(gold | GG) × P(GG) / P(gold) = 1 × (1/3) / (1/2) = 2/3
• Count the coins: 3 gold coins could have been drawn; 2 are in GG box, 1 in GS → 2 of 3 → 2/3

Did You Know?

🎩Joseph Bertrand posed this in 1889 in 'Calcul des probabilités'Source: Wikipedia

🚪The Monty Hall problem (1975) is mathematically equivalent — switching wins 2/3Source: Scientific American

🧠Even math professors initially answer 1/2 — base rate neglectSource: Cognitive psychology

📺Marilyn vos Savant's correct Monty Hall answer was attacked by thousands of PhDsSource: Parade Magazine, 1990

How It Works

1. The Setup

3 boxes: GG (2 gold), SS (2 silver), GS (1 gold + 1 silver). Pick a box at random, draw one coin.

2. The Observation

You drew gold. This eliminates Box 2 (all silver). What is P(other coin is gold)?

3. Counting Outcomes

3 gold coins could have been drawn. Two are in GG (other gold), one in GS (other silver). So 2 of 3 → P = 2/3.

4. Bayesian Proof

P(GG|gold) = P(gold|GG)×P(GG)/P(gold) = 1×(1/3)/(1/2) = 2/3. Bayes' theorem confirms.

Expert Tips

Count the coins, not the boxes

Focus on which specific coins could have been drawn

Simulate to verify

Run 10,000+ trials — watch it converge to 2/3

Connect to Monty Hall

Switching doors = same 2/3 logic

Use Bayes' theorem

Prior P(GG)=1/3, Likelihood P(gold|GG)=1, Evidence P(gold)=1/2

Comparison: Bertrand vs Related Paradoxes

Aspect	Bertrand's Box	Monty Hall	Boy/Girl Paradox
Correct answer	2/3	2/3 (switch)	1/2 or 1/3
Key insight	Count coins	Host reveals info	Order matters
Common mistake	Ignore GG bias	Ignore host info	Assume 50/50

Frequently Asked Questions

Why isn't the answer 1/2?

The all-gold box has two gold coins — when you draw gold, you're twice as likely to have picked from the GG box. So 2 of 3 gold draws come from GG.

How is this related to Monty Hall?

Both use conditional probability. In Monty Hall, switching wins 2/3 because the host's reveal gives information. Here, drawing gold favors the GG box.