Two people in a class share a birthday: solved step by step

Problem. What is the chance that at least two people in a group share a birthday? Assume 365 equally likely birthdays, ignore 29 February and twins, and treat each person's birthday as independent of the others.

Step by step

1. Count the opposite event instead: all n people have different birthdays. It is easier to calculate, and the chance of at least one shared birthday is 1 − (chance all are different).

2. The first person can have any birthday, so the chance is 365/365 = 1. The second person must avoid that date, which has probability 364/365. The third must avoid two dates, so 363/365.

3. Multiply these chances: P(all different) = (365/365) × (364/365) × ... × ((365 − n + 1)/365).

4. For n = 23 the last factor is 343/365. The whole product is about 0.493.

5. So the chance that at least two of 23 people share a birthday is 1 − 0.493 = 0.507, or 50.7%.

6. For a class of 30 the same product is about 0.294, so the chance of a shared birthday is about 70.6%.

7. For 50 people the chance rises to about 97.0%.

8. For 70 people it reaches about 99.9%.

Answer: With 23 people the chance that two share a birthday is 1 − 0.493 = 50.7%; it is about 70.6% for 30 people and 99.9% for 70 people.

Real birthdays are not perfectly uniform (some months are more common) and 29 February exists, so actual probabilities differ slightly from this model.