Tools
The birthday paradox
Ask people how many strangers it takes for a shared birthday to be likely and most say "a lot." The real answer is 23, for a coin-flip chance, and 70 for near-certainty. Move the slider and watch your intuition lose.
Chance at least two share a birthday
50%
Why your gut is wrong
The trick is that you are not comparing yourself to everyone; you are comparing every pair. With 23 people there are 253 possible pairs, and it only takes one match. Our intuition counts people and forgets the pairs explode quadratically. It is a small, humbling reminder that the mind is not built for combinatorics, the same theme running under exponential intuition.