Birthday Paradox Calculator

What is the Birthday Paradox?

The Birthday Paradox (also known as the birthday problem) is a famous counterintuitive result in probability theory. It states that in a room of just 23 randomly chosen people, there is a greater than 50% probability that at least two people share the exact same birthday.

Why is it Counterintuitive?

Our intuition compares a single individual's birthday to everyone else's (requiring 183 people for a 50% chance). However, the problem considers all possible pairs of people in the room. In a group of 23 people, there are:

$$\binom{23}{2} = \frac{23 \times 22}{2} = 253 \text{ pairs}$$

With 253 pairs of people, the probability of at least one shared birthday quickly exceeds 50%.

Mathematical Formula

The easiest way to calculate the probability of at least one shared birthday is to calculate the complementary probability that all $n$ people have distinct birthdays:

$$\bar{P}(n) = 1 \times \left(1 - \frac{1}{365}\right) \times \left(1 - \frac{2}{365}\right) \times \dots \times \left(1 - \frac{n-1}{365}\right)$$

The probability of at least two people sharing a birthday is then:

$$P(\text{shared}) = 1 - \bar{P}(n)$$

Key Benchmarks (365-day year)

• 23 people: 50.73% probability
• 41 people: 90.32% probability
• 57 people: 99.01% probability
• 70 people: 99.92% probability

Related Tools

Explore related tools to assist with your calculations:

• Birthday Calculator
• Random Birthday Generator