Revisiting The Birthday Paradox

Charlottesville

August 18, 2025

A while ago I wrote about the birthday paradox. To remind, the question is how many people do you need to have greater than a 50 percent probability that two of the people share the same birthday.

This is an example of a veridical paradox the definition of which is “a statement or situation that, despite appearing contradictory or absurd, is actually true.”

It's a paradox where the conclusion, while surprising or counterintuitive, is logically sound and can be demonstrated to be correct.

Key characteristics of veridical paradoxes:

Counterintuitive conclusion:

The result or statement challenges our initial expectations or common sense understanding.

True conclusion:

Despite the counterintuitive nature, the conclusion is logically sound and can be proven true.

Not a contradiction:

The paradox arises not from a logical flaw, but from a surprising or unexpected outcome based on accepted principles.

Examples of veridical paradoxes:

Monty Hall Problem:

This probability puzzle involves a game show scenario where a contestant chooses a door, and the host reveals a "zonk" behind another door. The paradox arises from the counterintuitive result that the contestant should switch their choice to increase their chances of winning.

Birthday Paradox:

In a group of people, the probability that at least two share the same birthday increases surprisingly quickly as the number of people grows. Even with just 23 people, the probability is greater than 50%.

Bertrand's Box Paradox:

This probability problem involves calculating the probability of drawing a gold coin from a box, given that a randomly drawn coin from one of the three boxes is gold. The result is a higher probability than intuition might suggest.

Zeno's Paradoxes:

These paradoxes, particularly the Achilles and the Tortoise, explore the nature of motion and infinity, seemingly demonstrating that movement is impossible, but the conclusions are actually incorrect, not a contradiction.

In essence, veridical paradoxes highlight the difference between our intuitive understanding and logical truths. They demonstrate that what seems impossible or contradictory can sometimes be both true and logically sound.

These type of problems fascinate me ever since I read Daniel Kahneman’s “Thinking Fast and Slow.” Kahneman opened my eyes to the fact that our mind’s tend to be lazy and rely too much on intuitive thinking.

Back to the birthday paradox.

One of the keys to understanding the solution is to understand how many pairs exist for a given number. To work backward from the answer (which is 23), there are 253 possible pairs that can be formed from 23 distinct numbers. This is calculated using the combination formula: n! / (r! (n-r)!), where n is the total number of items (23 in this case), and r is the size of the pairs (2). So, 23! / (2! 21!) = (23 * 22) / 2 = 253.

All of a sudden (at least for me), the answer of 23 seems “less unreasonable” than my intuitive reaction was.

So want to know how to solve the birthday paradox? Go to the 5:40 mark of this video.