1. What Is the Birthday Paradox?

The birthday paradox demonstrates that in a group of just 23 people, there's a 50% chance that two people share the same birthday. This seems counterintuitive, hence the term "paradox," but it's mathematically sound.

2. How the Calculation Works

The calculator uses the birthday paradox approximation formula:

Where:

• \( P \) — Probability of shared birthday (0 to 1)
• \( n \) — Number of people in the group
• \( e \) — Euler's number (~2.71828)
• 365 — Days in a year (can be adjusted)

Explanation: The formula calculates the probability that no two people share a birthday and subtracts this from 1 to get the probability of at least one shared birthday.

3. Practical Applications

Details: The birthday paradox has applications in cryptography, hashing algorithms, and understanding probability in everyday situations. It helps illustrate how quickly probabilities can rise with increasing group sizes.

4. Using the Calculator

Tips: Enter the number of people in your group. You can also adjust the number of days in a year (default 365) for different scenarios (like birthdays within the same month).

5. Frequently Asked Questions (FAQ)

Q1: Why is it called a paradox?
A: It's called a paradox because the probability is much higher than most people intuitively expect for small group sizes.

Q2: How many people are needed for 50% probability?
A: Only 23 people are needed for a 50% chance of a shared birthday with 365 days in a year.

Q3: How many people for 99% probability?
A: About 57 people are needed for a 99% probability of a shared birthday.

Q4: Does this account for leap years?
A: The calculator uses 365 days by default, but you can adjust this to 365.25 or 366 if you want to account for leap years.

Q5: Can this be used for other shared dates?
A: Yes, you can adjust the number of days for other scenarios (e.g., same birth month with 12 days).