Bertrand's Box Paradox Calculator

Explore the classic probability puzzle of Bertrand's Box Paradox and calculate conditional probabilities to understand this fascinating statistical problem.

Calculate Your Bertrand's Box Paradox Calculator

Box 1

🟡 🟡

Two Gold Coins

Box 2

⚪ ⚪

Two Silver Coins

Box 3

🟡 ⚪

One Gold & One Silver Coin

What is Bertrand's Box Paradox?

Bertrand's Box Paradox is a classic probability puzzle introduced by French mathematician Joseph Bertrand in his 1889 book "Calcul des Probabilités." The paradox demonstrates how conditional probability can be counterintuitive, leading many people to an incorrect solution based on faulty reasoning.

The Puzzle Statement

You have three identical boxes:

  • Box 1 contains two gold coins
  • Box 2 contains two silver coins
  • Box 3 contains one gold coin and one silver coin

You randomly select a box, and then randomly draw one coin from it. The coin turns out to be gold. What is the probability that the other coin in the box is also gold?

The Common Incorrect Answer: 1/2

Many people reason as follows: Since I've drawn a gold coin, the box must be either Box 1 (two gold coins) or Box 3 (one gold and one silver). Since there are only two possibilities for the box, and only one of them (Box 1) has another gold coin, the probability must be 1/2.

This reasoning is incorrect because it fails to account for the fact that you're more likely to draw a gold coin from Box 1 than from Box 3, which affects the conditional probabilities.

The Correct Solution: 2/3

The correct way to solve this problem is to use Bayes' Theorem. Let's define the events:

  • G: The drawn coin is gold
  • B₁: Box 1 was selected (two gold coins)
  • B₂: Box 2 was selected (two silver coins)
  • B₃: Box 3 was selected (one gold and one silver)

We want to find P(B₁|G), the probability that Box 1 was selected given that a gold coin was drawn. Using Bayes' Theorem:

P(B₁|G) = [P(G|B₁) × P(B₁)] / P(G)

Where:

  • P(G|B₁) = 1 (probability of drawing a gold coin from Box 1)
  • P(B₁) = 1/3 (prior probability of selecting Box 1)
  • P(G) = P(G|B₁)P(B₁) + P(G|B₂)P(B₂) + P(G|B₃)P(B₃)
  • P(G) = 1 × (1/3) + 0 × (1/3) + (1/2) × (1/3) = 1/3 + 0 + 1/6 = 1/2

P(B₁|G) = (1 × 1/3) / (1/2) = (1/3) / (1/2) = 2/3

Therefore, the probability that the other coin is also gold is 2/3 or approximately 67%.

Understanding the Solution

To gain intuition for why the answer is 2/3 rather than 1/2, consider the following:

  • Out of the 3 boxes, there are a total of 6 coins: 2 gold in Box 1, 2 silver in Box 2, and 1 gold + 1 silver in Box 3.
  • Of the 6 coins, 3 are gold (2 in Box 1 and 1 in Box 3).
  • If you draw a gold coin, it must be one of these 3 gold coins. 2 of them are from Box 1, and 1 is from Box 3.
  • Therefore, there's a 2/3 chance that the gold coin came from Box 1, meaning there's a 2/3 chance that the other coin in the box is also gold.

Visual Explanation with Coin Pairs

Another way to understand this is to list all possible pairs of coins that could be in the selected box, and which coin of the pair was drawn:

BoxCoins in BoxCoin DrawnOther Coin
Box 1Gold, GoldGold (1st)Gold
Box 1Gold, GoldGold (2nd)Gold
Box 2Silver, SilverSilver (1st)Silver
Box 2Silver, SilverSilver (2nd)Silver
Box 3Gold, SilverGoldSilver
Box 3Gold, SilverSilverGold

Of these 6 equally likely scenarios, 3 involve drawing a gold coin. In 2 of those 3 scenarios, the other coin is gold. Therefore, the probability is 2/3.

Why This Paradox Matters

Bertrand's Box Paradox illustrates several important concepts in probability:

  • Conditional Probability: It demonstrates how our intuitions about probability can be wrong when dealing with conditional events.
  • Bayes' Theorem: It provides a simple example of how to apply Bayes' Theorem to update probabilities based on new information.
  • Sample Space Analysis: It shows the importance of correctly identifying the sample space of equally likely outcomes.

Understanding this paradox has practical applications in many fields where conditional probability is important, including medicine (diagnostic testing), forensic science, data analysis, and decision making under uncertainty.

See Also

Frequently Asked Questions

Bertrand's Box Paradox is a probability puzzle introduced by Joseph Bertrand in his 1889 book 'Calcul des Probabilités.' It involves three boxes: one containing two gold coins, one containing two silver coins, and one containing one gold and one silver coin. After randomly selecting a box and drawing one coin (which turns out to be gold), the paradox asks for the probability that the other coin in the same box is also gold.

The correct answer is 2/3 or approximately 0.67. Many people incorrectly answer 1/2, thinking there are two possibilities for the other coin (gold or silver) with equal probability. However, proper application of conditional probability shows that when a gold coin is drawn, it's more likely to have come from the box with two gold coins than from the mixed box.

Using Bayes' Theorem: Let G be 'the drawn coin is gold' and B₁, B₂, B₃ represent the three boxes. We want P(B₁|G), the probability we selected the gold-gold box given we drew a gold coin. By Bayes' Theorem: P(B₁|G) = [P(G|B₁)×P(B₁)]/P(G) = [1×(1/3)]/[(1×1/3)+(1/2×1/3)+(0×1/3)] = (1/3)/(1/3+1/6) = (1/3)/(1/2) = 2/3.

People often get the wrong answer because they fail to properly account for the conditional probabilities involved. The intuitive answer of 1/2 neglects the fact that drawing a gold coin provides information about which box was likely selected. This is a common cognitive bias when dealing with conditional probability problems.

Bertrand's Box Paradox highlights the importance of carefully applying conditional probability principles, particularly Bayes' Theorem. It demonstrates how our intuitions about probability can be misleading and emphasizes the need for rigorous mathematical analysis in probability problems where new information becomes available.

Understanding this paradox has practical applications in fields requiring conditional probability reasoning, such as medical diagnostics, forensic science, data analysis, and decision theory. It illustrates the importance of properly updating probabilities based on new evidence, which is fundamental to Bayesian statistics and rational decision-making.

These are two different probability puzzles introduced by Joseph Bertrand. Bertrand's Box Paradox involves selecting boxes and coins, demonstrating conditional probability. Bertrand's Paradox (about random chords in a circle) illustrates how different definitions of 'random' can lead to different probability outcomes for seemingly the same question.

Yes, the solution can be verified through simulation or physical experimentation. If you were to repeat the experiment many times (selecting a random box, drawing a coin, and checking the other coin when the first is gold), you'd find that in approximately 2/3 of cases, the other coin is also gold, confirming the mathematical solution.

Both Bertrand's Box Paradox and the Monty Hall problem demonstrate how conditional probability can be counterintuitive. In both cases, new information (drawing a gold coin or having a door revealed) changes the probability landscape in ways that many people find surprising. Both illustrate the importance of properly applying Bayes' Theorem to update probabilities.

Variations include changing the number of boxes, altering the contents of the boxes, drawing multiple coins, or revealing information about specific boxes. These variations can be solved using the same conditional probability principles but may lead to different probabilities and provide additional insights into the nature of probability updating.

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