What is the probability that at least two out of 25 randomly selected students share the same birthday?

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To solve this problem, we can use the concept known as the Birthday Paradox. This paradox demonstrates that in a relatively small group of people, the odds of two individuals having the same birthday can be surprisingly high.

The approach involves calculating the probability that no two students share a birthday and then subtracting that from 1 to find the probability that at least two students do share a birthday.

Assuming there are 365 days in a year (ignoring leap years for simplicity), the probability that the first student has a unique birthday is 1, as they can have any day. The second student will have 364 out of 365 days available to avoid sharing a birthday with the first student. The third student will then have 363 days to maintain unique birthdays, and so forth.

The formula for the probability that all 25 students have different birthdays can be expressed mathematically as:

P(n) =   rac{365!}{(365-n)! imes 365^n}

For our case with 25 students (n=25), this becomes:

P(25) =   rac{365!}{(365-25)! imes 365^{25}}

Calculating this directly can be cumbersome, but for practical purposes, we can derive it as:

P(25) ext{ (all unique)} = rac{365 imes 364 imes 363 imes … imes 341}{365^{25}}

Now, we compute this step-by-step:

  1. For the first student: Probability of unique birthday = 1 (365/365)
  2. For the second student: Probability of unique birthday = 364/365
  3. For the third student: Probability of unique birthday = 363/365
  4. Continue this pattern up to the 25th student.

Multiplying these probabilities together gives us:

P( ext{all different}) = 1 imes rac{364}{365} imes rac{363}{365} imes … imes rac{341}{365} ≈ 0.4313

Now, to find the probability that at least two students share a birthday:

P( ext{at least one shared}) = 1 – P( ext{all different})

Substituting our previous calculation:

P( ext{at least one shared}) ≈ 1 – 0.4313 ≈ 0.5687

This means there is about a 56.87% probability that at least two students among 25 randomly chosen students will share the same birthday. This result is often surprising to many, as it illustrates how quickly probabilities can accumulate in smaller groups.

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