The problem you mentioned is known as the birthday paradox or birthday problem. The counterintuitive nature of the problem arises due to the assumption that birthdays are uniformly distributed throughout the year.

To calculate the probability, we can use the concept of complementary probability. That means we will calculate the probability that no two people share the same birthday and subtract it from 1 to find the probability that at least two people do share the same birthday.

Let's assume there are n people in the group. The first person can have any birthday with a probability of 1 (since there are no restrictions yet). The second person must have a different birthday than the first person, which leaves 364 possible birthdays out of 365 days. The third person must have a different birthday than the first two, or there are 363 possible birthdays out of 365 days, and so on.

Therefore, the probability that no two people have the same birthday in a group of n people is given by:

P(n) = (365/365) * (364/365) * (363/365) * ... * [(365 - n + 1)/365]

To find the probability that at least two people share the same birthday, we can subtract this probability from 1:

P(at least 2 people share a birthday) = 1 - P(n)

Let's calculate the probabilities for different group sizes:

- For n = 2 (two people in the group):
P(at least 2 people share a birthday) = 1 - (365/365) * (364/365) ≈ 0.003 ≈ 0.3%

- For n = 23 (typical classroom size):
P(at least 2 people share a birthday) = 1 - (365/365) * (364/365) * (363/365) * ... * (343/365) ≈ 0.507 ≈ 50.7%

- For n = 30 (larger group size):
P(at least 2 people share a birthday) ≈ 0.706 ≈ 70.6%

As you can see, with just 23 people, the probability of at least two people sharing a birthday exceeds 50%. This probability increases rapidly as the group size increases.

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