What is the birthday paradox? And is it real?

If I invite a group of people to a party, how likely are some of them to share the same date of birth? The birthday paradox is a mathematical phenomenon that shows the probability of having two people in a group with the same date of birth. The result may amaze you. In this article we show this paradox and explore this wonderful mathematical concept.

What is the birthday paradox? And why is it surprising?

The birthday paradox is an amazing concept in probability that shows how likely two people are to participate in a group on the same date of birth. In a group of just 23 people, there's a greater chance of at least two people sharing the same birthday. Of course, this possibility increases rapidly as each additional person is added to the group. The "paradox" comes from the fact that the result is very counterintuitive, because most people would guess that the probability should be much lower, i.e. we would need a much larger number of 23 people to achieve it. In fact, it may seem at first glance that we will need at least half the number of days of the year (about 182 people) to have a 50% chance of a joint birthday, because there is a 1/365 chance that someone else will have the same date of birth as you. What confirms our intuition is that you've met many more than 23 people, but you don't know anyone who shares your birthday (or you know but very few). How can this be true?

Solve the puzzle:

Surprisingly, the number 23 is perfectly correct, and it can be proven mathematically. The key is that our simple intuition ignores the fact that we have to not only check the date of birth of one person compared to others, but also check all possible pairs of people in the group. In other words, we're not just comparing someone's birthday to everyone else's birthday, but we're comparing everyone in the group to all the other people in it. With 23 people, there are many different couples of people whose dates of birth we can compare.

For example, imagine that the group consists of only two people: you and another person, only one pair. There is actually a small possibility (1/365) that you both have the same day of birth. Now that a third person is added to the group, there are three pairs to compare: you and the second person who was with you, you and the third person who was added, and the second person with the third. As more people are added, the number of couples will increase a lot, so the chances of two people sharing the same birthday also increase. By the time you have 23 people in the group, there are 253 couples to compare, and it is this large number of comparisons that makes the probability reach about 50%.

Full mathematical answer:

To better understand the problem, let's first determine the situation mathematically. Let's say there are 365 days a year, and we ignore leap years for the sake of simplicity. We want to calculate the probability that in a group of n people, at least two share the same date of birth. Each person's birthday is supposed to be equal on any of the 365 days, and birthdays are independent of each other. It's easier to calculate the supplementary probability (i.e., the probability that no one will participate on the date of birth), and then subtract this from 1 to get the probability that at least two people will participate on the date of birth.

First - the case of one person: The first person can have any birthday, so the probability that their date of birth is unique is 1 (i.e. 100%).

Case of two persons: In order for the date of birth of the second person to be different from the date of birth of the first person, he has 364 options (out of 365), since the first person has already taken a day. The probability is 364/365.

Third: For a third person to have a different date of birth than the first two, they have 363 days to choose from. The probability is 363/365.

Pattern persistence: When we add more people, each new person has one less day available to choose from to avoid sharing their birthday with the rest of the group. For the fourth person: 362 / 365, for the fifth person: 361 / 365 and so forth...

To calculate the probability that two people on the same date of birth do not participate in a group of n people, we multiply all of these probabilities together. This gives the probability that each new person will have a different birthday than everyone else who has preceded them. So, for n people, the probability of no common dates of birth is: (364/365). ( 363 / 365). ( 362 / 365)..... (365-N+1/365).

For example, when n = 23, the probability that two people will not share the date of birth is

(364 / 365). ( 363 / 365). ( 362 / 365)..... (343 / 365)

We can calculate this finding using a simple calculator, and the result is approximately 0.4927 (about 49.27%). Now, to find the probability that at least two people share a date of birth, we subtract this value from 1 and find 1−0.4927=0.5073, so the probability is about 50.73% for 23 people.

What when a group is more than 23?

Of course, the probability will increase rapidly: with 30 people (e.g. a classroom), the probability jumps to about 70%. With 50 people (e.g. a train car passenger), the probability is about 97%. With 70 people (people invited to your party, for example), the figure is over 99.9%, which means that two people are almost certain to share the same Christmas Day in a group of 70 people.

The Christmas paradox is a famous example in probability theory, showing how relatively small groups can produce surprisingly high probabilities for certain outcomes (such as shared birthdays). It's a great example of how human intuition often struggles with possibilities, and how our intuition can mislead us, especially when it comes to multiple comparisons. It also clearly shows the flawed logic that we can sometimes use.