# Birthday Paradox Calculator

URL: https://speedycalculator.com/calculators/statistics/birthday-paradox/
Last updated: 2026-10-25

> The birthday paradox calculator finds the probability that at least two people in a group share a birthday. With just 23 people the chance is over 50%.

## Inputs

- Number of people
- Days in year

## Results

- Chance of a shared birthday
- Number of pairs

## Formula

```
P(shared) = 1 − 365! ÷ ((365 − n)! × 365ⁿ)
```

## Explanation

The result surprises people because the number of possible pairs grows quickly: 23 people form 253 pairs.

### How to use the Birthday Paradox Calculator

- **Number of people**
- **Days in year**
- Want to work backwards? Type a value into **Chance of a shared birthday** and **Number of people** is calculated for you.

Results update as you type, and every field accepts the units shown in its menu.

### How it works

The calculator computes the chance that all birthdays differ using log-gamma, then subtracts from 1.

### Inputs explained

FieldWhat to enterNumber of peopleA numberDays in yearA number

### Example values

How the main result changes when **Number of people** changes and everything else stays at the example values above:

Number of peopleChance of a shared birthday (%)11.515.38817.2532.2942350.7328.7567.44534.580.5014694.825

## Worked example

Inputs — Number of people: 23; Days in year: 365.

Results — **Chance of a shared birthday = 50.73 %**; Number of pairs = 253.

## FAQ

**Does this ignore leap years?**
Yes, and it assumes birthdays are equally likely, which is close enough.

**How many people for 99%?**
About 57.

**What formula does the Birthday Paradox Calculator use?**
P(shared) = 1 − 365! ÷ ((365 − n)! × 365ⁿ).

**Can I work backwards from the result?**
Yes. Type the result you want into Chance of a shared birthday and the calculator finds the matching number of people.