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 make 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. I suspect it is 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 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 another door, the Teacher gives you two pieces of new information.”

“What information 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? The door where I’m standing may have nothing, or may have the coin. The other door may have nothing, or may have the coin. The odds are still the same!”

“No, my Son. When she opened that empty door, your Teacher most likely had no choice. That’s the second piece of information.”

“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 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 that I’m thinking about it, our Teacher doesn’t really have a choice. I’m standing behind one door with nothing, so there is only one left.”

“And the other door?”

“The other door has the coin, Master.”

“Exactly. So the odds are two out of three that you’re standing behind an empty door, in which case your Teacher has no choice but to open the one remaining empty door, because the other door has the coin. Thus, by switching to that door, 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 purpose and its limitations. That is what your Teacher was trying to convey to you.

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. 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, an hourglass on the side. They sat in quiet contemplation, thinking about their next moves. 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?”

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