Shared Birthdays

Solution & Explanation

The problem you're dealing with is a classic probability puzzle known as the Birthday Paradox. Despite its name, it's not really a paradox but rather a counter-intuitive result in probability theory.

Problem Statement:

You need to calculate the probability that in a group of k people, at least two individuals share the same birthday.

Assumptions:

1. There are 365 days in a year.
2. Each person's birthday is equally likely to be any of the 365 days.
3. Birthdays are independent of each other.
4. Leap years, twins, and other variations are disregarded.

Steps to Solve:

1. Calculate the Complement Probability:

   • Instead of directly calculating the probability of at least two people sharing a birthday, it's easier to calculate the probability of the complementary event: no two people share the same birthday.

2. Probability of No Shared Birthdays:

   • For the first person, any of the 365 days are possible.

   • For the second person, 364 days are available to avoid matching the first person's birthday.

   • For the third person, 363 days are available, and so on.

   • The probability that all k people have different birthdays is:

     $P_{\text{no shared}}(k) = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \ldots \times \frac{365-k+1}{365}$

3. Simplify the Expression:

   • This can be simplified using factorial notation:

     $P_{\text{no shared}}(k) = \frac{365!}{(365-k)! \times 365^k}$

4. Calculate the Desired Probability:

   • The probability that at least two people share a birthday is the complement of the above probability:

     $P(k) = 1 - P_{\text{no shared}}(k)$

Example Calculations:

• For k = 2:

  • $P_{\text{no shared}}(2) = \frac{365}{365} \times \frac{364}{365} \approx 0.99726$
  • $P(2) = 1 - 0.99726 \approx 0.00274$

• For k = 23:

  • The probability gets surprisingly high:
  • $P(23) \approx 0.5073$
  • This means there's about a 50% chance that in a group of 23 people, at least two share a birthday.

Conclusion:

The Birthday Paradox demonstrates how our intuition about probability can often be misleading. Even with a relatively small group of people, the probability of shared birthdays becomes significant due to the large number of potential pairings.