When Should We Expect to Find Gold?

Exploring the Bertrand’s Box Dilemma

A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

Leighton is on BlueSky (leightonvw.bsky.social), Threads (leightonvw), and Twitter (@leightonvw).

THE GAME AND THE PUZZLE

The Bertrand’s Box Paradox, first posed by mathematician Joseph Bertrand, offers a fascinating challenge to our intuitive grasp of probability.

In Bertrand’s scenario, there are three indistinguishable boxes. Each is closed. The first box contains two gold coins, while the second box holds two silver coins. The third box contains one gold and one silver coin. This setup paves the way for an exploration of probability and decision-making that might seem to challenge common sense.

THE CHOICE AND THE IMPLICATION

Imagine yourself in this setting. You randomly select one of these boxes and, without looking, you take one of the two coins from that box. As you open your hand, you see a shiny gold coin resting in your palm.

Now, the presence of the gold coin means that you didn’t select the box containing the two silver coins. Thus, the box in front of you must either be the one containing two gold coins or else it is the one containing one gold and one silver coin. With this information, what is the probability that the other coin in the box is also gold?

THE INTUITIVE ANSWER AND THE SURPRISE

At first glance, the problem appears simple. Having excluded the box containing the two silver coins, we are left with two possible boxes: a box with two gold coins, and a box with one gold and one silver coin. Based on this information, we might presume that the likelihood of each box being the one we randomly selected should be equal. This presumption would lead us to the intuitive conclusion that the chance the other coin is gold stands at 1/2. Likewise, the chance that it is silver would also be 1/2. But is this intuition correct?

In fact, the truth diverges from this intuitive explanation. The correct answer to the probability that the other coin is gold is not 1/2, but 2/3. This outcome might seem to defy common sense. How could merely examining one coin influence the composition of the remaining concealed coin?

THE REVELATION AND THE TRUE ANSWER

To solve this puzzle, we need to look deeper into the details. To do so, let’s imagine that each coin in the boxes has a unique label. In the gold coin box, we have Gold Coin 1 and Gold Coin 2. In the mixed box, there’s Gold Coin 3 and Silver Coin 3, while the silver box holds Silver Coin 1 and Silver Coin 2.

When we initially drew a gold coin from our chosen box, three equally likely events could have occurred. We could have drawn Gold Coin 1, Gold Coin 2, or Gold Coin 3. We remain unaware of which specific gold coin we hold, but the outcomes for the remaining coin in the box vary based on this choice. If we had picked Gold Coin 1 or Gold Coin 2, the remaining coin in the box would also be gold. So, there are two chances it would be gold. However, if it was Gold Coin 3, the other coin in the box would be silver. This is one chance compared to the two chances it is gold.

When we consider these equally likely scenarios, the probability that the other coin is gold stands at 2/3, whereas the probability that it’s silver is 1/3. A seemingly simple choice of coin selection reveals in this way a solution that seems to challenge our intuitive understanding of probability.

THE IMPACT OF NEW INFORMATION

Before we drew the gold coin, the probability that we had chosen the box with two gold coins was 1/3. But when we uncovered the gold coin, we didn’t merely exclude the box with two silver coins, we also gathered new information. Specifically, we could have drawn a silver coin if our selected box was the one with mixed coins, yet we drew a gold coin. This fresh piece of information now means that it is twice as likely that we chose the box with two gold coins rather than the mixed one, because there were two ways this could have happened, compared to just one way if we had selected the mixed box.

A BRIEF SUMMARY OF THE BOX PARADOX

Imagine there are three boxes:

Box 1 contains two gold coins.

Box 2 contains one gold coin and one silver coin.

Box 3 contains two silver coins.

Initially, without any further information, the probability of choosing any one of the boxes is 1/3.

When you draw a coin and see that it is gold, you’re effectively eliminating Box 3 (the box with two silver coins) from consideration because it cannot possibly be the box you chose. This leaves you with Box 1 and Box 2 as possibilities.

However, the key insight is in how we update our probabilities based on the new information that the drawn coin is gold:

For Box 1 (with two gold coins), there are two chances of drawing a gold coin (since both coins are gold).

For Box 2 (with one gold and one silver coin), there is only one chance of drawing a gold coin.

Therefore, given that you have drawn a gold coin, it is twice as likely that you have chosen Box 1 compared to Box 2. This updates the probabilities to:

2/3 chance that the box chosen was Box 1 (with two gold coins).

1/3 chance that the box chosen was Box 2 (with one gold and one silver coin).

CONCLUSION: THE KEY LESSON

The paradox of Bertrand’s Box serves to remind us of the nuanced nature of probability, and illustrates the central importance of incorporating new information into probability calculations. Ultimately, though, it highlights the deceptive character of common intuition, encouraging us to challenge our feelings with the power of reason.