Want to find someone with the same birthday as you? It’s not that difficult!

When you hear someone has the same birthday as you, will you exclaim "what a coincidence" or even unconsciously develop a sense of closeness towards him/her? Is it God's will that you were born on the same day and met each other in the vast sea of ​​people?

After scientific calculations, we have to say that this idea is too emotional. After all, the probability of two people being born on the same day may be much greater than you think.

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1

What is the probability of having the same birthday in a class?

Suppose there are 40 students in a class in an elementary school. What is the probability that they have the same birthday (same month and same day)?

This is actually a permutation and combination problem. First, assuming that people are born on the same day, there are many possible combinations besides the simplest one - two people born on the same day. There are cases where people have the same birthday on different dates, such as two people born on March 14 and two people born on April 13. There may be more than two people born on the same day, for example, there are three people born on March 14.

If we consider this, there may be complex situations such as three people born on one day and four people born on another day. If we want to list every possible combination and add up the probabilities, it is actually an almost impossible task.

However, if you think about it from the opposite side, the problem becomes much simpler.

The sum of the probabilities of having duplicate birthdays and not having duplicate birthdays in the same class is 1. We just need to calculate the probability of not having duplicate birthdays and then subtract this probability from 1 to get the conclusion we want.

Thus, we can simplify the problem to the probability that no two (or more) people in an elementary school class of 40 are born on the same day.

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For convenience, we assume that everyone is invited out of the classroom first, and then call the students back one by one, and in the process calculate the probability that the birthdays of the new students are different from those of the previous students.

Suppose the birthday of the first student who enters the classroom is March 14th. We invite the second student to enter. To meet the requirements of the question, the second student's birthday can be any day of the 365 days except March 14th. The probability that the birthday is different from the first student's is 364/365. (Here we make two assumptions. The first is that leap years are not considered, and the second is that the birth rate should be equal every day throughout the year. )

Please invite the third student to enter. His birthday cannot be the same as the previous two students. So now the probability becomes (364/365)×(363/365). The first bracket is the probability that the first two students have different birthdays, and the second bracket is the probability that the third student has a different birthday from the first two. The result of multiplication is the probability that all three students have different birthdays. The probability that four people have different birthdays is (364/365)×(363/365)×(362/365)...

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And so on, ** continue calculating until the 40th person, and then subtract the calculated probability from 1, which is the answer to the question we want to know, **that is, the probability of repeated birthdays among 40 people.

The final result is 89.1%. Is it larger than expected?

If the number of students continues to increase, the probability will rise sharply. For a class of 50 people, the probability is 97.0%, for 60 people, it reaches 99.4%, and for 70 people, it is already 99.9%. In other words, the probability that no one in a class of 70 people has the same birthday is less than one in a thousand.

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Tip: In the actual process, we don’t need to calculate stupidly thirty or forty times. Computer software (a simple spreadsheet will do) can help us complete this repetitive and tedious task.

There is a very classic mathematical "paradox" called the "birthday problem": How many people are needed in a room so that the probability that two of them have the same birthday is greater than 50%?

According to the calculation method above, we can easily get the answer, 23 people, I believe this number is less than most people's intuitive estimate. Although it is called a "paradox", the birthday problem is not a paradox from the perspective of causing logical contradictions. It is called a paradox only because this mathematical fact contradicts general intuition. After all, most people would think that the probability of two people out of 23 having the same birthday should be much less than 50%.

2

What are the chances of meeting someone with the same birthday as you?

At this point, you may have a question: since the probabilities calculated above are unexpectedly large, why have I never met someone who was born on the same day as me in my class since I was a child?

In fact, if you are smart enough, you should realize that this is another proposition - what is the probability of having someone with the same birthday as you in a class of 40 people?

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We still use the same thinking method as inviting students into the classroom one by one to solve the problem. First calculate the probability that no one in a class of 40 has the same birthday as you, and then subtract this value from 1 to get the result we need.

First, “I” enters the classroom. The probability that the second student enters the classroom has a different birthday from “me” is 364/365. The probability that the second and third students have different birthdays from “me” is (364/365)×(364/365). When the fourth student enters the classroom, the answer is (364/365)×(364/365)×(364/365)…

Similarly, when entering the nth classmate, the probability is (364/365) to the power of n-1. Finally, we subtract the above result from 1, which is the probability of having someone with the same birthday as you in a class of n people. The calculation results are as follows: a class of 4 people (0.8%), a class of 23 people (5.8%), a class of 40 people (10.1%)...

The result is more in line with our general understanding than the previous question. So in a class of 40 people, the probability of having a classmate with the same birthday as you is 10.1%.

Each of us will join many classes from childhood to adulthood. From the above calculation results, it is a real miracle if there are no people with the same birthday in any class from childhood to adulthood! We calculated that there are 60 people in each elementary school class, 70 people in each junior high school class, 50 people in each high school class, and 30 people in each university class. The result is less than 5 in 10 million, which is at the level of lottery jackpot in terms of probability.

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Therefore, the probability of having the same birthday in a group of people is much greater than many people expect, not to mention the billions of people around the world.

Of course, since there is no significant difference in the birth rate every day, the total population on a certain date (note that the date is not a specific year plus date, such as March 14, not March 14, 1985) is about 20 million among the 7 billion people in the world. If we consider the people who have died in history, the number of people born on a certain day must be astronomical, and countless celebrities are born or die on any day.

So, although we hope that every day is a beautiful, special, and magical day, in fact, every day is ordinary and common, and no day can be called a "miracle day."

This article is produced by Science Popularization China, produced by Li Rui (Osaka University), and supervised by the Computer Network Information Center of the Chinese Academy of Sciences

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