Someone told me that it was disgusting to insert a formula at the beginning of the article, so I stopped writing the calculation process (besides the traditional permutation and combination method, American mathematician paul halmos also gave a clever solution) and gave the result directly. The probability of 50 people in the same Amanome is as high as 97%, which is probably far beyond most people's expectations.
Believe me, the mistake is not the calculation process, but the intuition. In this place, science once again played a joke on our daily experience. It is precisely because of the obvious contradiction between the calculation results and daily experience that this problem is called "birthday paradox". It embodies the contradiction between rational calculation and perceptual knowledge, and does not cause logical contradiction, so it is not a paradox in a strict sense. The original expression of "Birthday Paradox" is that the probability of 23 people sharing the same Amanome is more than 50%. In order to make the contradiction more prominent, I deliberately changed the number to 50. If you don't know the answer in advance and just look at the question, your guess must be far below 97%, right? Having said that, some people may question that when we calculate, we assume that people's birthdays are evenly and randomly distributed, but this is not necessarily the case in life-don't worry, mathematicians have long considered this factor, and the uneven distribution has been solved, which further proves that the probability will only be higher when it is unevenly distributed. In addition, American computer scientist D. E. Knuth once calculated such a question: How many people can find the same Amanome on average? The answer is 25 people. Does this look incredible?
Why is there such a contradiction? Actually, the problem is not complicated. First of all, when there is only 1 person, the probability of "the same Amanome" is 0%. If leap years are not considered, when the number of people is more than 365, the probability of "the same Amanome" is 100%. Therefore, in the range of 1 to 365, we usually naturally think that the corresponding probability increases linearly from 0% to 100%, even if it is not linear, it will not be too steep, so for 50 people, the probability should be about 50/365, that is, 13.7%. But in fact, the growth momentum of this curve is very terrible: it rises linearly from the beginning, and by the time 50 people arrive, the probability is close to 100%, which is very different from the straight curve we imagined. So the question is: Why do we mistakenly think it is linear? Don't worry, you can get inspiration by changing the question a little. The new question is, what is the probability that someone in a group is in the same Amanome as you? Similarly, when we draw the probability curve, we can see that it is very gentle: when the number of people is 350, the probability is only slightly higher than 50%.
Now, we can roughly find the reason for the false intuition mentioned above: when we see that "someone is the same as Amanome", we subconsciously speculate with "it's the same as my birthday", so that the rocket launch is regarded as a steady growth, resulting in the "birthday paradox".
Calculation method
The probability of the first person is one in 365, the probability of the second person is two in 365, the probability of the third person is three in 365, the probability of the fourth person is four in 365 ... The probability of the fiftieth person is 50 in 365.
Then multiply them all to get p (identical) = 0.03 and p (identical) = 1-0.03 = 0.97.
0.97=97‰
Have the same probability of 97‰