The crowded room
People walk into a room one at a time, each with a random birthday. How many will walk in before two of them share one? First, make a guess.
0people in the room
0%chance of a shared birthday by now
–average room size at first match (0 rooms)
The trick is that you're not asking whether someone shares your birthday. You're asking whether any two people share one. With 23 people there are 253 different pairs, and each pair is another chance for a match.
With just 23 people in the room, a shared birthday is more likely than not. With 70, it's almost certain.