Bertrand's Box Paradox

 I came across the following problem in my Inbox:

I have placed three coins inside a bag. The first coin has heads on both of its sides, the second coin has tails on both of its sides, and the third coin has heads on one side and tails on the other side.

I randomly take one coin from the bag. I open my hand and look at it. I notice that the side facing me is heads. What is the probability that the other side of the coin is also heads?

A. Less than 50%, B. More than 50% or C. Exactly 50%

I went for the exactly 50% option, which unfortunately proved to wrong. The correct answer was more than 50% and the following explanation was offered:

This puzzle is a variant of the Bertrand’s box paradox, and the exact answer is that there is a 2/3rds chance that the other side of the coin is also heads. However, many people answering this question intuitively say there is a 50% chance.

The best way to think of this problem is in terms of sides, not individual coins that have two sides. We are told that there are three heads sides in total. The heads side that you viewed could be any of the three heads sides in the bag. Two of the three heads sides in the bag have heads on the other side, therefore there is a 2/3rds chance that the other side of the coin you viewed is also heads.

I was thus prompted to find out more about Bertrand's Box Paradox and this is what Gemini had to say:

Bertrand's Box Paradox is a classic problem in probability theory that often trips people up because their intuition leads them to the wrong answer. It's a "veridical paradox," meaning the solution is indeed counter-intuitive but demonstrably correct.

Here's the setup:

Imagine you have three identical boxes:

1. Box 1 (GG): Contains two gold coins.

2. Box 2 (SS): Contains two silver coins.

3. Box 3 (GS): Contains one gold coin and one silver coin.

You randomly choose one of the three boxes. Then, you randomly draw one coin from that chosen box.

The Paradoxical Question:

If the coin you drew is gold, what is the probability that the other coin remaining in that same box is also gold?

Why it's confusing (the intuitive, but wrong, answer):

Many people would intuitively say 1/2 (50%). Their reasoning often goes like this:

"Since I drew a gold coin, I know I didn't pick Box 2 (SS). So, it must be either Box 1 (GG) or Box 3 (GS). Since there are only two possibilities, and one of them (GG) means the other coin is gold, it's 1/2."

Why this reasoning is flawed:

The flaw lies in assuming that Box 1 (GG) and Box 3 (GS) are equally likely given that you drew a gold coin. This is not true.

The Correct Solution (and why it's 2/3):

Let's break down the possibilities more systematically. There are a total of 6 coins (2 gold, 2 silver, 1 gold, 1 silver). Each coin has an equal chance of being drawn initially.

Consider all the ways you could draw a gold coin:

1. From Box 1 (GG): You could draw the first gold coin, and the other coin is gold. (Let's call this G1a)

2. From Box 1 (GG): You could draw the second gold coin, and the other coin is gold. (Let's call this G1b)

3. From Box 3 (GS): You could draw the gold coin, and the other coin is silver. (Let's call this G3)

These are the only three ways you could have drawn a gold coin. Each of these three "initial gold coin draws" is equally likely.

Now, let's look at what the other coin in the box would be for each of these three equally likely scenarios:

1. If you drew G1a (from Box 1), the other coin is gold.

2. If you drew G1b (from Box 1), the other coin is gold.

3. If you drew G3 (from Box 3), the other coin is silver.

Out of the three equally likely ways to draw a gold coin, two of those ways result in the other coin also being gold.

Therefore, the probability that the other coin in the box is also gold is 2/3.

Analogy to help understand:

Imagine you have six coins laid out on a table:

• G1 G2 (from Box 1)

• S1 S2 (from Box 2)

• G3 S3 (from Box 3)

You close your eyes and pick one coin. You open your hand and it's gold. Which gold coin did you pick? It could be G1, G2, or G3.

• If you picked G1, the other coin in its pair (from Box 1) is G2 (gold).

• If you picked G2, the other coin in its pair (from Box 1) is G1 (gold).

• If you picked G3, the other coin in its pair (from Box 3) is S3 (silver).

Since G1, G2, and G3 were all equally likely to be picked as your first gold coin, and two out of those three scenarios lead to the other coin being gold, the probability is 2/3.

The key takeaway is that when you draw a gold coin, you gain information that changes the probabilities of which box you initially chose. The box with two gold coins is twice as likely to yield a gold coin on the first draw compared to the box with one gold and one silver coin.