The Birthday Paradox

The room fills one person at a time, each new arrival only able to dodge the birthdays already taken, until the odds flip. This lesson works out exactly where they flip, and why the number is nothing like the one your gut reaches for.
Before you start. All the arithmetic here is counting: how many people, how many pairs they make between them, nothing harder than that. Every number below is worked out in one plain sentence the moment it shows up, so nothing arrives bare.
