1. What is the Birthday Paradox?

The Birthday Paradox demonstrates that in a group of just 23 people, there's a 50% chance that at least two people share the same birthday. This seems counterintuitive given there are 365 days in a year.

2. How Does the Calculator Work?

The calculator uses the probability formula:

Where:

• \( P \) — Probability of at least one shared birthday
• \( n \) — Number of people in the group
• \( ! \) — Factorial operation

Explanation: The formula calculates the complement probability (that all birthdays are unique) and subtracts it from 1.

3. Understanding the Results

Key Points: The probability rises rapidly with group size. At 23 people it's 50.7%, at 41 people it's 90.3%, and at 70 people it's 99.9%.

4. Using the Calculator

Tips: Enter any integer between 2 and 365. The calculator will show the probability that at least two people share a birthday in that group.

5. Frequently Asked Questions (FAQ)

Q1: Why is this called a paradox?
A: It's called a paradox because the result seems counterintuitive - people often underestimate how quickly the probability rises with group size.

Q2: Does this account for leap years?
A: No, this calculation assumes 365 equally likely birthdays. Adding February 29th would slightly decrease the probabilities.

Q3: What about real-world birthday distributions?
A: Actual birthday distributions are slightly uneven, which would increase the probabilities slightly compared to the theoretical calculation.

Q4: How is this useful in real life?
A: It's important in probability theory, cryptography (birthday attacks), and understanding coincidences in everyday life.

Q5: Why does the probability rise so quickly?
A: Because the number of possible pairs grows quadratically with group size (23 people have 253 possible pairs).