Is the birthday paradox correct? Why the birthday paradox is wrong with reality

Can be very responsible to tell you, exactly right.

This question is assigned by our python teacher in class, I have written a program in python to simulate this problem, and you can see the result, if it is a class of 50 people, after 100,000 sample class simulations, 97142 samples have the same, and the probability birthday is generated by counting computer random numbers).

Now let's explain it from probability theory:

There are n people, and the first birthday is 365 to 365

The second person is 365 to 364 (if the second person is going to have a different birthday than the first).

The third person is 365 to 363 (which is different from the first and second people).

The nth person is 365-n+1 (different from the previous one).

So all people are different, that is: (yes are not the same).

365/365）*（364/365）*.365-n+1 365) (backslash is a semicolon).

Having the same is 1 - the above equation (the opposite of not being the same is the same, even if only two are the same).

Substituting n=50, we can get that the probability is about what I calculated with a Casio scientific counter, which is basically the same as my python simulation results).

The birthday paradox says that there are 23 people, and the probability of two people having the same birthday will be more than 50% (similar to the above, I verified that it is greater than 50% through scientific counters and python programs).

Birthday paradox. If there are 23 or more people in a room, there is a greater than 50% probability that at least two people will have the same birthday. This means that in a typical standard elementary school class (30 students), there is a higher probability that two people will have the same birthday.

For people 60 or more, the probability is greater than 99%. The birthday paradox is not a paradox in the sense that it contradicts general intuition. Most people would think that the probability that 2 out of 23 people have the same birthday should be much less than 50%.

What are the odds of two out of 23 people having the same birthday?

More than 50%!

This means that it is not uncommon for two people in a class to have the same birthday.

Let's use mathematical knowledge to explain this problem.

Taking a common year as an example, calculate the probability that all the birthdays in the room are not the same, then:

The first person's birthday is 365 out of 365;

The second person's birthday is 365 out of 364;

The third person's birthday is 365 out of 363

The nth person's birthday is 365 out of 365-(n-1).

So the probability that everyone's birthday is not the same is:

Then, the probability that at least two people in n people have the same birthday is:

So when n=23, the probability is .

When n=100, the probability is .

The following is calculated using random variables:

Let x[i,j] represent the probability that the birthdays of the ith person and the jth person are different, then it is easy to know that any x[i,j]=364 365

Let event A indicate that n people have different birthdays.

Solution p(a)<1 2, logarithm: n>=23

In contrast, random variables are just as simple to understand and much easier to calculate.

Intuitively illustrating this problem, it is actually necessary to understand that the combination of the same birthday can be quite a lot.

As mentioned in the previous example, 23 people can produce 23 22 2 = 253 different combinations, each of which has an equal chance of success. From this point of view, it is not so incredible to produce a successful pair out of 253 combinations.

Finally, put a thinking easter egg.

There are as many dots in a 1 cm segment as there are dots on the surface of the Pacific Ocean"?

Cantor (1845-1918) succeeded in proving:

A point on a straight line can correspond to a point on a plane, and it can also correspond to a point in space.

Due to infinity, there are "as many points" in a 1 centimeter-long line segment as there are points on the Pacific Ocean, and as many points in the interior of the entire Earth.

Can you get to the dots?

End-Series: Fawn's Math Forest.

A lot of things in the world aren't black and white. There are many things that cannot be pushed in detail, but a trade-off will produce contradictions, which is often referred to as a paradox. There are still many questions about paradoxes.

There are also people who wonder if the tortoise and hare race is a paradox, the magical Fermi paradox, and so on. This is another magical paradox – the birthday paradox.

1. The birthday paradox.

This means that there are 23 or more people in a room, so there is a greater than 50% chance that at least two people have birthdays on the same day. This also means that in a primary school class of 30 people, there is a greater likelihood that two people will have the same birthday. If the number of people is several times 30, the probability will be above 99%.

Although this does not seem to be a paradox from the point of view of causing logical contradictions, it can only be called a paradox in the sense that this mathematical fact conflicts with general intuition.

Second, the content of the paradox.

If there are 23 or more people in a room, there is a greater than 50% chance that at least two people will have birthdays on the same day. This means that in a typical standard elementary school class (30 students), there is a higher probability that two people will have the same birthday. For people over 60 years old, the probability is greater than 99%.

There are no special years and months, such as leap February.

First calculate the probability that everyone in the room has a different birthday, and then the first person's birthday is 365 vs. 365. The second person's birthday is 365 to 364. The third person's birthday is 365 to 363.

The nth person's birthday is 365 in 365-(n-1). So when n=23, the probability is. When n=100, the probability is.

For individuals who have already been identified, the probability of different birthdays varies. Calculations were performed using the following random variables:

Let x[i,j] represent the probability that the i-th person and the j-th person have different birthdays, then it is easy to know that any x[i,j]=364 365.

Let event a indicate that n people have different birthdays. In contrast, random variables are just as easy to understand and much easier to calculate.

3. Understand paradoxes.

The crux of the matter is to recognize that matches for the same birthday can be many. For example, 23 people can produce 23 22 2 = 253 different combinations, and each of these combinations has an equal probability of success. From this point of view, it is not so incredible to produce a winning pair out of 253 combinations.

On the other hand, if you enter a room with 22 people, the probability that everyone in the room will have the same birthday as you is not 50%, but becomes very low. The reason for this is that only 22 different combinations can be produced at this point. The birthday question actually asks what is the probability that two out of 23 people will have the same birthday.

Mainly for the reason of probability, 23 people can produce 23 22 2 = 253 different pairs, and each of these pairs has an equal probability of success, so the probability of two of the 23 people having the same birthday is more than 50%.

"An important reason is that when n is getting larger, the magnitude of the change in the value is much smaller than the change in the value of n. For example, n is equal to 10 when it is equal to 100, and n is equal to 100 when it is equal to 10 000. Substituting 365 into this formula, we get:

Of course, there is no possibility of a human being in reality, but this means that as long as the sample size exceeds this value, the probability of having the same birthday will be more than 50%. This formula proves that if the sample size is 23 people, then the probability must be more than 50%. This formula has a very wide range of applications, which is very convenient for us to make similar calculations.

The first main reason is that there is a moment of reunion at this hour, and the second point is that time is very magical, once things meet, then there must be a chance of success.

The main reason is that there are too many people in the world, so this probability will increase.

Summary. Hello dear, I'm glad to answer why the birthday paradox is not the same as the actual error: the birthday paradox is a probability fallacy that states that in a random set of 23 or more people, there is a greater than 50% probability that at least two people will have the same birthday.

The reason why this paradox is related to actual error is that people often underestimate this probability. In real life, people tend to think that it takes more people to be able to have the same birthday, but in reality, only 23 people are enough for the paradox to hold. This error may come from our intuition understanding of probability and statistics, emphasizing empirical feelings and ignoring mathematical laws.

There are also birthday attack apps.

Hello dear, I'm glad to answer why the birthday paradox is not the same as the actual error: the birthday paradox is a probability fallacy that states that in a random set of 23 or more people, there is a greater than 50% probability that at least two people will have the same birthday. The reason why this paradox is related to actual error is that people often underestimate this probability.

In real life, people tend to think that it takes more Zen infiltrators to be able to have the same birthday, but in reality, it only takes 23 people to make the paradox true. This error may come from our lack of intuitive understanding of probability and statistics, emphasizing the experience of wisdom and ignoring the laws of mathematics.

Hello, I'm glad to answer the question of how a birthday attack should be closed: a birthday attack is a cryptographic attack method, the basic idea of which is to use the birthday paradox to find collisions in hash functions. In a hash function, messages of any length are mapped to a digest (hash value) of a fixed length, and different messages should have different hash values.

However, due to the paradox of the generation of segments, if the hash value and the number of bits of the message are equal, then it is possible to find two messages with the same hash value by randomly generating a considerable number of messages. Therefore, birthday attacks can be used to crack passwords or tamper with digital signatures by constructing fake messages with the same hash value. Birthday attacks are common in the field of cybersecurity and are widely used in cryptography.

For example, an attacker can use a birthday attack to forge a digital certificate to carry out a man-in-the-middle attack, steal bank account information, and more. Therefore, when designing a cryptosystem, it is necessary to consider the threat of birthday attacks, and use more secure hashing algorithms or methods such as increasing the length of the hash value to defend against birthday attacks.

Birthday paradox. It means that the probability that two out of 23 people have the same birthday date can be reached, which is almost more than half of the total number of people, and according to our daily thinking, this probability is considered completely impossible, so it is called a paradox.

But in fact, the correct probability that it can be calculated is not a mathematical paradox.

In fact, the reason why people have a very low probability that 23 people have the same birthday is mainly because most people are standing under a fixed thinking premise, that is, we instinctively fix 23 people as people in the same room, but in fact, we can see it as a random 23 people, so that the collocation between them is far more than 22, but as many as 253, then the chance of the same probability will be greater.

First of all, we want to know what is the probability that two people in any 23 people have the same birthday, then we can achieve it by inference, first of all, the probability that the first person in the 23 people, he and the other people's birthdays are not the same is 365 365, because everyone's birthday may be one of the 365 days in a year, and the probability that the second birthday is not the same is 364 365, and so on, the probability that the birthday of the 23rd person is not the same is 343 365.

And these are equivalent to the independent probability that each of them has a different birthday, to calculate the common probability, we need to multiply these fractions, that is, 365 365*364 365... 343 365, the final result is that this is the probability that two people's birthdays are not the same, then the reverse is the probability that two people have the same birthday, you need to subtract 1 to get it, and it is concluded that the probability that two people have the same birthday in any group of not less than 23 people is always more than half.

And according to this algorithm, it can be obtained, with the continuous increase of the number of people in the room, the probability of at least two of them having the same birthday will be higher and higher, such as when reaching 30 people, then the probability of two people having the same birthday will reach 70%, and when there are 70 people, the probability of two people having the same birthday is as high as possible, which can be said to be close to affirmation, of course, there are many paradoxes that are different from common sense, such as the liar's paradox.