It's intuitive if you do it visually. Connect each person to every other person with a piece of string. 10 people each have about 10 connections. 50 people each have about 50 connections. 1000 people each have about 1000 connections.
When you're handing out 250 pieces of string it's easy to tell that something with a 1/365 chance per string is likely.
For this situation, if you put 50 thousand people in a room then they each have 50 thousand ways to pair up, so you're looking at over a billion opportunities to have a one-in-a-billion coincidence. (50k * 50k / 2 = 1.25 billion)
When you're handing out 250 pieces of string it's easy to tell that something with a 1/365 chance per string is likely.
For this situation, if you put 50 thousand people in a room then they each have 50 thousand ways to pair up, so you're looking at over a billion opportunities to have a one-in-a-billion coincidence. (50k * 50k / 2 = 1.25 billion)