Zen and the Three-Door Problem

A student came to see a Master, and said “Master, I am confused. My class is learning Probability Theory, and I am already struggling with our very first assignment.”

“Why don’t you tell me more,” said the Master.

"Well, Master, our Teacher asked us to imagine ourselves standing in front of three closed doors. Behind one of the doors she has hidden a gold coin. Behind the other two doors is nothing. We are invited to pick one of the three doors. If we pick the right door, the coin is ours.

After we’ve made our choice, the Teacher will open another door, with nothing behind it of course. Then, she will give us a chance to switch doors if we want to.

Now the question is, should we stay put or should we switch doors?"

“And what do you think is the right answer?”

“To stay put, Master. It’s a trick question. The Teacher is probably just testing our steadfastness.”

“No, my son. She is helping you.”

“How is she helping us, Master?”

“Let’s discuss the odds. At first, when you chose a door, your chances of picking the door with the coin were…?”

“One out of three, Master,” said the student.

“That’s right,” said the Master. “You had no reason to prefer any one door over another. But by opening that other door, the Teacher gives you a valuable clue.”

“What would that be, Master?”

“Well, the door that she is opening leads to…?”

“To nothing, Master. She knows where the coin is hidden. But how does that help? There are still two doors left to choose from. Whether I move or stay put, the odds are the same!”

“No, my son. When she opened that empty door, your Teacher most likely had no choice. That is the clue you must follow.”

“Why would our Teacher have no choice, Master?”

“Well, where does the door where you’re standing most likely lead to?”

“To nothing, Master. Two doors have nothing, one has the coin.”

“Exactly. And if so, where do the remaining doors lead to?”

“One to nothing, one to the coin, Master.”

“So, how many doors can your Teacher choose from?”

The student thought for a moment.

“Well, now I’m thinking about it, it is like you said, Master. Our Teacher has no choice. If I’m standing at a door with nothing, there is only one door that she can open. The other one has the coin.”

“Exactly. So the odds are two out of three that you’re standing behind an empty door. In that case, once your Teacher has opened the other door, the remaining door has the coin. Thus, by switching doors, your odds of winning become two out of three.”

“Whereas they remain one out of three if I stay put. Now I understand.”

The student stayed quiet for a while, and the Master said: “You still seem troubled, my son.”

“Yes, Master. I am troubled because I was misguided by my intuition. What good is our intuition if we can’t trust it?”

"We can trust our intuition, my son, as long as we understand its limitations.

Imagine walking through a forest and hearing a twig snap, somewhere in the distance. What would your intuition tell you?"

“To run for my life, Master.”

“Which would probably be a good idea. And how much time would you need for deliberation?”

“No time at all, Master. I would be running before I knew.”

“Indeed, my son. Our intuition works very quickly. It trades accuracy for speed. It tells you what to do when you can’t afford to wait. Given that, when would a deliberate decision be better?”

The student thought for a moment.

Under a nearby tree, two monks sat on opposite ends of a Shogi board, thinking about their next moves, an hourglass on the side. Eventually, one of them moved a piece on the board and turned the hourglass.

At that moment, the student understood.

He turned to the Master, and said: “When there is time to think?”

The Master nodded. “Yes, my son. When there is time to think, and when the decision matters.”

One of the monks looked up from the Shogi board and met his opponent’s eyes. Then he laid down his king, and smiled.

Further reading

• Wikipedia article on Bayes’ Theorem
• Wikipedia article about the three door (Monty Hall) problem
• Book Thinking, Fast and Slow by Daniel Kahnemann, ISBN 9780374533557