There are 365 days in a year and the probability of each person's birthday is 1/365. Suppose there are n people as the observed sample and the probability of 2 people having the same birthday is 100% when n is greater than 365. When 1 ≤ n ≤ 365, what is the probability that at least 2 people have the same birthday. The perceptual understanding of this problem is that the closer n is to 365 the greater the probability of occurrence, and the probability increases with n. Although the speed will be accelerated, it will not be outrageous. In fact, when n=3, the probability that at least 2 people have the same birthday is only 1%; The probability increases rapidly to 50.7% for n=23 and to 70.6% for n=30. The rate of probability improvement is exponentially related to the observed sample size. The problem is called the "birthday paradox" because of the obvious deviation of the computational results from the perceptual experience. It reflects the contradiction between rational calculation and perceptual understanding, not a logical contradiction.

Qiyang Sun. The probability principle of the birthday paradox and extended applications. The Frontiers of Society, Science and Technology (2021) Vol. 3, Issue 3: 110-112. https://doi.org/10.25236/FSST.2021.030320.

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