To find the probability that at least two out of 25 randomly selected students share the same birthday, we can use the complementary probability approach. First, we calculate the probability that no two students share a birthday and then subtract that from 1.
Assuming there are 365 days in a year (ignoring leap years), the probability that the first student has a unique birthday is 1 (or 365/365). For the second student to have a different birthday, there are 364 days available, making the probability for the second student 364/365. For the third student, the probability becomes 363/365, and so on.
The formula for the probability that all 25 students have different birthdays is:
P(no shared birthday) = 365/365 × 364/365 × 363/365 × … × (365 - 24)/365
This can be expressed mathematically as:
P(no shared birthday) = 365! / (365 - 25)! / 365^{25}
Calculating that:
- P(no shared birthday) ≈ 0.431
Now, to find the probability that at least two students share a birthday, we subtract this value from 1:
P(at least one shared birthday) = 1 – P(no shared birthday) ≈ 1 – 0.431 = 0.569
Therefore, the probability that at least two out of 25 randomly selected students share the same birthday is approximately 0.569 or 56.9%.