Learning path
Probability That Tricks Your Intuition
Five puzzles where the correct answer feels obviously wrong. Together they form a short course in the two questions people never think to ask: how many chances were there really, and how did I come to be shown this?
The Birthday Problem
The gentlest possible start. Nothing is hidden and nobody is tricking you — you simply counted people when you should have counted pairs.
In a room of just 23 people, there is a better than even chance that two of them share a birthday. Not 183 people. Twenty-three. The reason is that 23 people make 253 pairs, and every pair is a chance to match.
BeginnerCounterintuitive result14 min
Bertrand's Box Paradox
Now something is hidden. The information you receive depends on how it was produced, and this 1889 puzzle is where that lesson was first packaged.
Three boxes, two drawers each: two gold coins, two silver, one of each. You open a drawer and find gold. The chance the other drawer in that box is also gold is 2/3, not 1/2 — and this is exactly the Monty Hall problem, posed a century earlier.
BeginnerVeridical paradox10 min
The Monty Hall Problem
Bertrand's box with a television set and a goat. Famous because it is easy and people still get it wrong — including, reportedly, Paul Erdős.
You pick one of three doors. The host opens a different door to reveal a goat, then offers you the chance to switch. Switching doubles your odds of winning — and almost nobody believes it the first time.
BeginnerVeridical paradox9 min
The Boy or Girl Paradox
The turn. Here the honest answer is that the question was never precise enough to have one answer, which is a more useful thing to learn than any particular number.
A family has two children and at least one is a boy. What is the chance both are boys? Martin Gardner said 1/3 in 1959, then conceded it could equally be 1/2. The answer depends on how you learned the fact — and the puzzle never says.
IntermediateApparent contradiction13 min
Simpson's Paradox
Out of the puzzle books and into real data. Every group in a study can show one trend and the combined data the opposite, and this actually happened at Berkeley in 1973.
A treatment can work better for small stones, better for large stones, and still look worse overall. Every group prefers one thing and the total prefers the other. The arithmetic is correct both times.
IntermediateVeridical paradox20 min
Berkson's Paradox
The final form: the trend is not merely reversed, it was manufactured by the way the sample was collected. Once you can see this you cannot stop seeing it.
Two unrelated things can look strongly related once you only look at a selected slice. Among hospital patients, among the people you date, among everyone you hired — the selection itself invents the correlation.
IntermediateCounterintuitive result15 min
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