At the 2014 FIFA World Cup, exactly half of the 32 squads of 23 players had two players sharing a birthday, and five teams had two such pairs. That is the birthday problem in action: in a group of just 23 random people, there is a 50.7 percent chance of a shared birthday, rising to about 71 percent at 30 people and 99.9 percent at 70. This episode explains why that feels wrong and why it is true.

It walks through the inversion trick of calculating the odds that everyone stays unique, the 253 handshakes hidden inside a group of 23, and the ego trap that makes matching your own specific birthday require 253 people. It covers Harold Davenport's unpublished discovery and Richard von Mises' 1939 paper, why uneven birth distributions make matches more likely, the near-match variant needing only seven people, the coupon collector's problem requiri ...