Probability puzzles
Our gut is bad at chance. These five puzzles fool almost everyone the first time, including people with maths degrees. Make your guess first, then let your device run each one thousands of times to see what really happens.
Stay or switch?
On a TV game show, a new car is behind one of three doors and a goat is behind each of the other two. You pick a door. The host, who knows where the car is, opens another door to show a goat. Then he asks: do you want to switch to the last closed door?
Play it yourself
How many people for a shared birthday?
Picture a room of people: a class, a wedding hall, a family gathering. How many people do you need for a better than even chance that two of them share a birthday (the same day and month, not the same year)?
Try any group
Is tails "due"?
You flip a 1-dirham coin. It lands heads five times in a row. It's a fair coin, and you're about to flip it again.
Predict each flip
You test positive. Now what?
1 in 100 people have a certain disease. A screening test catches 90% of the people who have it, but it also wrongly flags 9% of healthy people. You take the test and it comes back positive.
Now change the numbers
Better in every group, worse overall?
A real 1986 study compared two treatments for kidney stones. Treatment A (open surgery) worked better for small stones. It also worked better for large stones.
Flip the view
Your gut against the maths
Your first guess on each puzzle, and what really happens.
Why chance fools us
In the birthday puzzle we think about people, but the maths is about pairs. In the test puzzle we think about the test, but the maths is about how rare the disease is.
Real randomness is clumpy. Long streaks and coincidences are normal, so we see patterns and "due" outcomes that aren't there.
When a hidden factor decides who ends up in which group, totals can point the wrong way. Ask how the numbers were collected before trusting them.
Turn percentages into people ("9 out of 100"), run the experiment many times, or draw it. Your gut is fast, but counting is right.
Sources: the Monty Hall problem as set out by statistician Steve Selvin (The American Statistician, 1975); the birthday problem by exact calculation, ignoring 29 February and assuming birthdays are spread evenly; longest runs of heads from Mark Schilling, "The Longest Run of Heads" (The College Mathematics Journal, 1990); the screening-test example and the study of doctors from Gerd Gigerenzer and colleagues, "Helping Doctors and Patients Make Sense of Health Statistics" (Psychological Science in the Public Interest, 2007); kidney stone data from Charig and colleagues (BMJ, 1986). Every other number on this page is simulated live on your device. The medical example is a maths puzzle, not medical advice: talk to your doctor about any real test result.