Real Kato Blog: Counter-Intuitive

A couple of critical thinking puzzles for you. Try not to hurt your think bone.

Over on kottke.org recently, there was a discussion about a puzzle that showed up on Straight Dope. The puzzle goes like this:

A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?

The answer to this is actually a little bit tricky, and requires a bit of thought. What's amazing is just how many people get the answer wrong even after it's been explained to them fully. You can read the train of comments over on kottke's post, and marvel at how many people keep suggesting the same wrong answer, for the same wrong reasons.

This reminded me of a similar controversy that erupted after Marilyn Vos Savant posted the Monty Hall problem, which goes like this:

A player is shown three closed doors; behind one is a car, and behind each of the other two is a goat. The player is allowed to open one door, and will win whatever is behind the door. However, after the player selects a door but before opening it, the game host (who knows what's behind the doors) must open another door, revealing a goat. The host then must offer the player an option to switch to the other closed door. Does switching improve the player's chance of winning the car?

The answer to this question is also counter-intuitive... and not only that, but hundreds of mathematics professors wrote to Marilyn that she was wrong. She was, in fact, absolutely correct. But again, even faced with a complete explanation of the correct answer, people continued to insist that she was wrong.

You can click the links for the explanation of the answers, or scroll down a bit for the short version.

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1. Yes, the plane can take off. The plane's wheels are not powered; they roll freely. Thus the conveyor belt's speed has almost no bearing on the airplane's speed, which is determined by how fast the engines pull the airplane through the air. The wheels will be turning twice as fast as they normally would on takeoff; assuming they don't explode, the plane should take off normally.

2. Yes, switching improves the odds, making switching a better choice 2/3 of the time. For me, it's just a matter of working out the probability tree, and the answer is irrefutable. For those of you who need a more intuitive answer, think of it this way: you are taking advantage of the host's knowledge of what's hidden behind each door to improve your odds. Suppose there were a thousand doors. You select one, then the host opens 998 others, leaving just your door and the winning door closed. Even though you're faced with a choice between just two doors, that's clearly not a 50-50 proposition, and you'd switch, right?

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