Updated Probability & Statistics Tool

Birthday Paradox Calculator

Explore the famous birthday paradox: how likely it is that at least two people in a group share a birthday. Compute the probability for any group size, or find the minimum group size needed to reach a target probability, for both 365-day and 366-day years.

Shared Birthday Probability Minimum Group Size 365 or 366 Day Year Probability Table

Calculate Shared Birthday Probabilities and Required Group Sizes

This Birthday Paradox Calculator has two complementary modes. In the first tab, you enter a group size n and see the probability that at least two people share a birthday, along with the probability that all birthdays are distinct and a table for smaller group sizes. In the second tab, you set a target probability and the calculator finds the smallest group size that meets or exceeds it.

The classic birthday paradox assumes equally likely birthdays and ignores seasonal effects. You can choose a 365-day year or a 366-day year (leap-year model) to examine the effect of including February 29.

Table shows probabilities for group sizes from 2 up to this value.

The probability that at least two people share a birthday is 1 minus the probability that all birthdays are distinct. For n > days per year, the shared-birthday probability is 1 (P = 100%) because of the pigeonhole principle.

Enter as a decimal between 0 and 1, for example 0.5 for 50%.
The calculator searches from n = 2 up to this value for the first group size meeting the target.

This mode finds the smallest group size n such that the probability of at least one shared birthday is greater than or equal to the target probability P. If no such group size is found up to the maximum search limit, the calculator reports that the limit was reached.

Birthday Paradox Calculator – Understanding Shared Birthday Probabilities

The birthday paradox is a famous probability puzzle: in a group of just 23 people, the chance that at least two of them share a birthday is already a little over 50%. That result feels surprising, which is why it is often called a “paradox,” even though the underlying math is straightforward.

The Birthday Paradox Calculator on MyTimeCalculator lets you explore this effect in detail. You can compute the shared-birthday probability for any group size n, generate a probability table for smaller groups, or work in reverse to find the minimum group size needed to reach a target probability such as 50%, 75% or 99%.

1. Core Idea of the Birthday Paradox

Instead of counting pairs that match directly, the standard birthday paradox derivation starts from the easier quantity: the probability that all birthdays in the group are distinct. For a year with d equally likely days (usually d = 365), the probability that n people all have different birthdays is:

P(all distinct) = 1 × &frac{d - 1}{d} × &frac{d - 2}{d} × … × &frac{d - (n - 1)}{d}.

The first person can have any birthday. The second person must avoid that day, the third must avoid the first two days, and so on. The probability that at least one shared birthday occurs is then:

P(shared) = 1 − P(all distinct).

2. How the Birthday Paradox Calculator Works

The calculator uses this product formula directly, updating it step by step to avoid unnecessary recomputation. For each person added to the group, the probability that all birthdays remain distinct is multiplied by another fraction. This is numerically stable and efficient for typical group sizes.

In the Probability for Group Size tab:

  • You enter the group size n and choose the number of days per year (365 or 366).
  • The calculator computes P(all distinct) and P(shared) = 1 − P(all distinct).
  • If n > d, the probability of a shared birthday is 1 by the pigeonhole principle.
  • A table shows how P(shared) grows as n increases from 2 up to a chosen cutoff.

In the Group Size for Target Probability tab:

  • You specify a target probability P (for example 0.5, 0.75 or 0.99).
  • The calculator searches increasing group sizes until P(shared) ≥ P.
  • It reports that minimum group size, along with the probability at n and at n − 1.

3. Classic Milestones in the Birthday Paradox

For a 365-day year and equally likely birthdays, some commonly cited milestones are:

  • n = 23: P(shared) just over 50%.
  • n = 30: P(shared) around 70%.
  • n = 50: P(shared) well above 90%.
  • n = 70: P(shared) above 99%.

The calculator reproduces these values and lets you experiment with other thresholds, such as 25% or 95%. You can also switch to a 366-day model to see how much including February 29 shifts the probabilities.

4. Assumptions Behind the Birthday Paradox

The classic birthday paradox uses a simplified model with the following assumptions:

  • All days of the year are equally likely as birthdays.
  • Birthdays are independent between different people.
  • The year has either 365 days (ignoring leap days) or 366 days (considering leap years).

In reality, birthday distributions are not perfectly uniform—there are seasonal patterns, and fewer births tend to occur on certain holidays. However, the uniform model is very close to reality for most purposes and captures the main effect driving the paradox.

5. How to Use the Birthday Paradox Calculator

  1. Decide what you want to know: the probability for a given group size, or the group size required for a given probability.
  2. If you know the group size, use the Probability for Group Size tab, enter n, choose 365 or 366 days per year and click the button.
  3. If you have a target probability (for example 0.5 or 0.99), use the Group Size for Target Probability tab and enter the probability as a decimal between 0 and 1.
  4. Optionally adjust the maximum group size for the search and the maximum n for the probability table.
  5. Read the summary text to see how the result compares to common rules of thumb and intuitive expectations.
  6. Experiment with different parameters to build intuition for how quickly P(shared) grows as n increases.

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Birthday Paradox Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about the birthday paradox, how the probabilities are calculated and how to interpret the results from this calculator.

It is called a paradox because the result feels counterintuitive, not because it is mathematically inconsistent. Many people guess that you would need a much larger group—close to 180 or more—to get a 50% chance of a shared birthday. In reality, just 23 people are enough in the 365-day model. The surprise comes from underestimating how quickly the number of possible pairs grows with group size.

Yes, you can switch between a 365-day year and a 366-day year using the “Days per year” dropdown in each tab. The 365-day option ignores February 29, which matches the most common classroom version of the puzzle. The 366-day option lets you see how the probabilities change when the extra day is included in the model.

No. In reality, some months have more births than others, and certain dates (such as major holidays) tend to have fewer births. However, the uniform model is close enough that it still gives a very good approximation for shared-birthday probabilities. In many real settings, clustering of birthdays can even make shared birthdays slightly more likely than the uniform model suggests rather than less likely.

The classic birthday paradox looks at the event that some pair of people share a birthday, not a specific date. If you want the probability that at least one person in a group has, for example, a birthday on January 1, the calculation is different: it is 1 minus the probability that nobody has that birthday. The Birthday Paradox Calculator focuses on the “any matching date” version, which is what makes the paradox so striking.

Yes, conceptually. The birthday paradox is a special case of a more general “collision” problem: you have a certain number of possible outcomes (days or hash values), and you draw items independently at random. The formulas used here are analogous to those used in analyzing hash collision probabilities in computer science. To adapt the idea, treat the number of hash values as the “days per year” and the number of items as the “group size.”

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