With the first person, you have 1/365 chance the birthday will be on any given day.
Each person you add to that adds not just another person but also another day that can be a match.
After two people, you still don’t have a match but now you have two days. The third person can match either of those. That’s a lower bar than person #2 had to meet.
After 15 people, the question is: “what are the odds that any two of you share any of these 15 days as your birthday.”
See how this already sounds a little more likely than just narrowing in on “have the same birthday?” When it’s said that way it sounds like more of a bullseye than it truly is.
So I think part of it is difficult to grasp intuitively, but it’s also phrased deliberately to throw off your intuition.
To grasp it intuitively, I think of it like this.
With the first person, you have 1/365 chance the birthday will be on any given day.
Each person you add to that adds not just another person but also another day that can be a match.
After two people, you still don’t have a match but now you have two days. The third person can match either of those. That’s a lower bar than person #2 had to meet.
After 15 people, the question is: “what are the odds that any two of you share any of these 15 days as your birthday.”
See how this already sounds a little more likely than just narrowing in on “have the same birthday?” When it’s said that way it sounds like more of a bullseye than it truly is.
So I think part of it is difficult to grasp intuitively, but it’s also phrased deliberately to throw off your intuition.
I can see it kinda. At the same time you are reducing the unique dates and increasing the people you could match with.