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 is mathematically correct.

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
• \( 365 \) — Number of possible birthdays (ignoring leap years)

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

3. Understanding the Results

Details: The result shows the probability that at least two people in a group of size n share the same birthday. For example:

• 23 people: ~50.7% chance
• 50 people: ~97% chance
• 70 people: ~99.9% chance

4. Using the Calculator

Tips: Enter the number of people in the group (1-365). The calculator will show the probability that at least two people share a birthday.

5. Frequently Asked Questions (FAQ)

Q1: Why is this called a paradox?
A: It's called a paradox because the probability is much higher than most people intuitively expect.

Q2: Does this account for leap years?
A: No, this calculation assumes 365 equally likely birthdays, ignoring February 29th.

Q3: What about real-world birthday distributions?
A: Actual birthday distributions are slightly uneven, which increases the probability further.

Q4: How is the calculation done for large groups?
A: The calculator uses logarithms to handle the very large numbers involved in factorial calculations.

Q5: What's the smallest group where probability reaches 100%?
A: By the pigeonhole principle, with 366 people, at least one shared birthday is guaranteed (100%).