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The birthday paradox: why 23 people are enough to share a birthday

Bet a friend that two people in a room of 23 share a birthday, and you win more often than you lose. Half of the teams at the 2014 football World Cup had such a pair. Learn the birthday paradox: why so few people are enough, how to count the chance with no formula, how many random picks any match needs, and the birthday attack that forgers base on it.