Bertrand's Box Paradox

Bertrand's box paradox is a probability problem that helps us understand how our perception of odds is easily mistaken! In this exhaustive article, you will learn:

• What is Bertrand's box paradox: the game's formulation and rules.
• Why the "common sense" result is wrong, and what the solution to Bertrand's box paradox is.
• The relationship between Bayes rule and Bertrand's box paradox.
• How to use our tool for the simulation of Bertrand's box paradox.

Are you ready? Pick your box!

Let's set up Bertrand's box paradox: start by gathering some gold and silver coins (or any other coins or objects belonging to two distinct types), and place them in three boxes, according to these rules:

• In the first box, place two gold coins.
• In the second box, place two silver coins.
• In the third box, place a gold coin and a silver coin.

Shuffle the boxes! The paradox is appropriately set up.

It's time to discover Bertrand's box paradox statement: If, after choosing a box, I randomly draw a coin from it, and it turns out to be gold, what is the probability that the other coin is golden, too? Don't even think of using the different money weight to alter your odds: just maths!

As there are two boxes containing gold coins, and if you draw a gold coin from one of them, the remaining one can be either a gold coin (if you chose the box with two gold coins) or a silver one (if you chose the box with a silver coin and a gold coin), it makes sense to expect the other coin to be gold with 50% of probability. Well, the answer is wrong!

Bertrand gave an explanation to his paradox that relied on showing why the "common sense" solution was wrong. His explanation remains the easiest to understand: imagine to pick your box, but then not draw the coin. Notice that the problem is symmetric; that is, you can switch gold for silver without changing the numbers!

Each gold coin has a probability $1/6$ of being drawn, but since we stop playing if the first extraction results in silver, we can consider only the three outcomes that result in gold at the first draw. Each of these outcomes has probability $1/3$ of happening.

If you chose the box with gold and silver, you have zero chance that after the extraction of the gold coin, the other will also result in a gold coin. This accounted for one of the three gold coins. The other two coins, if extracted, result in the other coin being gold too: as each of these outcomes has probability $1/3$, and we sum them, we get the correct solution to Bertrand's box paradox: the probability of finding a gold coin after the first extracted coin is gold is $2/3$.

After the paradox had been stated and solved, mathematicians applied conditional probability to Bertrand's box paradox: Bayes' rule allows us to find a solution to the problem just by stating the rules, without the need to think of the possible outcomes.

What is Bayes' rule? Bayes' rule is a valuable statistical tool that allows you to calculate conditional probability starting from a particular set of quantities. If you need a refresher on conditional probability, visit our conditional probability calculator. Here is the formula for Bayes' rule:

where:

• $P(B|A)$ — Posterior probability, the conditional probability of the event $B$ in the case the event $A$ happened.
• $P(A|B)$ — Likelihood, the conditional probability of event $A$ given the event $B$.
• $P(A)$ — Prior probability of event $A$, the event we consider as "happened" in the posterior probability.
• $P(B)$ — Normalization factor.

To adapt Bayes' rule to Bertrand's box paradox solution, we need to understand what are the quantities above in our situation.

• The posterior probability is the probability of finding a gold coin given that we already drew a gold one: this equals the probability of choosing the box with two gold coins given that we got gold in the first extraction. We call it $P(GG|G)$.

• The likelihood is the probability of drawing gold from the box that contains two gold coins. We call this probability $P(G|GG)$.

• The priori probability is simply the probability of choosing the box with two gold coins: $P(GG)$.

• The normalization factor is the probability of drawing gold: $P(G)$.

Let's assign values to these quantities:

• $P(G|GG)=1$: this result comes from the fact that if you pick the box with two gold coins, you are sure you'll extract a gold coin!
• $P(GG)=1/3$, as every box has the same probability of being chosen.

For $P(G)$, we need to think a bit more! Its value depends on the contributions from each box. Hence, the chance of finding a gold coin in every box are:

• In the box with two gold coins, we have probability $1$;
• In the box with a gold coin and a silver coin, we have probability $1/2$; and
• In the box with two silver coins, we have probability $0$.

We multiply these values by the probability of choosing each box and sum the results:

Substitute these values in Bayes' formula:

We found the solution to Bertrand's paradox!

Our tool can run a simulation of Bertrand's box paradox. Follow the instructions to see an explanation of the paradox and its solution, and change your choices to see how you can affect the outcome of the game!

Bertrand's box paradox is a statistical problem with a seemingly counterintuitive answer. The paradox is usually presented in this way:

1. Take three boxes, and put two gold coins in one, two silver coins in another, and a silver coin and a gold one in the last.
2. Choose a box, and draw only a coin from it: the other remains hidden.
3. If you picked a gold coin, what is the probability that the other coin in the box is gold, too?

The answer to the question is 2/3 even though, at first glance, 1/2 sounds a better choice!

In Bertrand's box paradox, the probability of drawing a second gold coin is 2/3, and not 1/2, because the coins in the box with two gold coins are independent in the calculations to find the probability of drawing another gold coin. There are three possible first extractions of a gold coin, but only two result in another one being extracted on the second turn!

You can use Bayes' rules for Bertrand's box paradox if you consider the following:

• The probability of finding a gold coin after drawing gold as the posterior probability P(GG|G).
• The probability of drawing gold from the box that contains two gold coins as the likelihood P(G|GG) = 1.
• The probability of choosing the box with two gold coins as the priori probability P(GG) = 1/3.
• The probability of drawing gold as the normalization factor P(G) = 1/2.

Use the following formula to find the solution:

P(GG|G) = P(G|GG) · P(GG)/P(G) = (1/3)/(1/2) = 2/3

The solution to Bertrand's box paradox is 2/3. This quantity corresponds to the probability of drawing gold a second time from a box containing two coins if the first extraction already resulted in gold. This result is related to the chances of choosing a box containing two gold coins rather than a box with a silver coin and a gold one or one with two silver coins; however, we add a degree of information due to the knowledge about a part of the content of the box (the first extraction).