Famous Puzzles Explained: Monty Hall, the Bat and Ball, and More

In 1990, Marilyn vos Savant, a columnist for Parade magazine, received thousands of angry letters for answering a puzzle correctly. Many came from people with PhDs who insisted she was wrong. The puzzle is called the Monty Hall problem, and it shows why famous puzzles earn their keep: they reveal exactly where our intuition breaks.

The answer to the Monty Hall problem

The setup comes from the game show Let's Make a Deal and its host, Monty Hall. There are three doors. Behind one is a car; behind the other two, goats. You pick a door. The host, who knows where the car is, opens another door that hides a goat and asks if you want to switch. Should you?

Yes: switching wins two times out of three. When you first chose, you had a 1/3 chance of being right and 2/3 of being wrong. The host opening a goat door does not change that first probability, and your door is still worth 1/3. But the entire remaining 2/3 now sits on the other closed door. What matters is that the host does not open a door at random: he always reveals a goat. Picture a hundred doors and it turns intuitive. You pick one, the host opens 98 goat doors, and offers to let you switch to the only other door he left shut. Would you stay with yours?

The bat and the ball

A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?

The answer that pops into your head is 10 cents, and it is wrong. If the ball cost $0.10 and the bat $1.00 more, the bat would be $1.10 and the total $1.20. The correct answer is 5 cents: the ball costs $0.05, the bat $1.05, and together they come to $1.10. This puzzle belongs to the Cognitive Reflection Test (CRT), which Shane Frederick published in 2005 in the Journal of Economic Perspectives. Frederick noticed that even students at highly selective universities often missed it, because they stuck with the intuitive answer without checking it.

Two more CRT puzzles that fool intuition

• The lily pads. A patch of lily pads on a lake doubles in size every day. If it takes 48 days to cover the whole lake, how long does it take to cover half? Intuition says 24, but the answer is 47, because on the last day the patch doubles and covers the entire lake.
• The machines. If 5 machines take 5 minutes to make 5 widgets, how long would 100 machines take to make 100 widgets? Five minutes: each machine makes one widget in 5 minutes.

The birthday paradox

How many people must be in a room for there to be better than a 50% chance that two share a birthday? Most people guess high, around 180. The correct answer is only 23 people, with a probability of about 50.7%. The reason is that we are not comparing your birthday with everyone else's, but everyone against everyone: with 23 people there are 253 possible pairs.

Why we miss these in particular

Because the brain likes the short road. When a question looks easy, we answer with the first idea and never check it. Puzzles are built so that the first idea fails, and in doing so they train something valuable: doubting the obvious for one second.

How to practice this kind of reasoning

A good habit is to take a puzzle and, before looking at the solution, write down your gut answer and how sure you are of it. Then compare. Over time you will notice which kinds of problems your intuition fails you on most: probabilities, percentages, exponential growth. Knowing that does not make you infallible, but it teaches you when to slow down and work it out on paper.

Explaining the solution out loud to someone else also helps. If you cannot explain it simply, you probably do not fully understand it yet, and that is exactly when you learn the most.

If you enjoyed watching intuition stumble, our test has logic questions where pausing for a second can change your score. Count how many times your first answer catches you out.