So maybe this isn't gambling, but let's pretend someone wagered, "I bet you 50 bucks that if I ask the next thirty people when their birthday is, none of them will have the same birthday." Do you take the bet?

Short answer, Yes.

But why? And why do people call this simple probability problem a paradox?

Humans are Bad at Probability

Let's start with what we're good at, reasonably close odds. Flipping a coin makes reasonable sense to us. It either happens or it does not, 50/50.

But what are the chances of a one in hundred year hurricane destroying your beach house? Well 1/100 obviously, but from a practical standpoint, you probably are not worried about it. You might only have 50 years to live, and then only a 50% of being bothered with it! But the storm will happen, eventually... This is a probability estimate we are bad at. We are even further at compounding events (like what the probability this storm won't happen in the next 5 years).

So back to our paradox. Meeting someone with your birthday has a 1/365 chance, which is very rare (even more so if you are born on February 29th). This might be the first figure to consider when posed the question. However, this is not the question. The question is 30 people. To incorrectly answer this, we could say well that is 30/365 days covered, so again, very remote chance anyone shares a date.

But that only considers sharing one person's birthday.

The correct way is to think of the problem as everyone walks in the door one by one. You are person 1. The next person to has a 1/365 of having your birthday. Assuming they don't have your birthday, the next person has a 2/365 of having either of your birthdays. The next person has 3/365 etc...

There is a compounding effect. What is the probability of person 2 not having your birthday and person 3 not having the same birthday as you or person 2? That is (364/365)*(363/365). Multiple 362/365 for the next person. Take one minus this product for the probability for not having the same birthday.

Another way to think about this is not just about asking the 30th person if they have any of the same birthdays as the 29 of you in the room, but also what is the probability no one before that had the same birthday as they walked in the door. If it was the 10th person (14% chance), you got to stop annoying people at the door quite a while ago. The 50% mark comes at the 22nd guest, or 23 people total.

Final Thoughts on this "Paradox"

Have some respect for the word paradox. Just because something is not intuitive to our monkey brains, doesn't make it a paradox. Soon paradox will have a little meaning as irony if we keep this up.

Next time someone asks you about the Birthday Paradox, shut them up and tell them yeah, "It is more likely than you think when you have a big group of people that at least two will share a birthdate. That's because with over about 23 people, there are enough possible combinations of pairs that its not that unlikely. But can god make a wall so high even (s)he could not jump over? If you keep halving a distance, will you ever reach the end? And did back to the future make sense, what happens if you kill your grandfather while time traveling?" That'll make you a hit at the party (or get the weird birthday math guy to stop talking to you)