The Birthday Problem | FractionRush

🎂 The Setup

Topics: probability, complement rule, multiplication rule

Question: In a room with n people, what is the probability that at least two of them share the same birthday?

Assume: 365 days in a year (ignore leap years), and birthdays are equally likely on any day.

Most people guess you'd need around 180 people for a 50% chance. The real answer is surprisingly small. How small?

Complement rule: It's easier to calculate P(no shared birthday) first, then subtract from 1.

P(at least one shared birthday) = 1 − P(all birthdays different)

P(all different, n people) = ³⁶⁵/₃₆₅ × ³⁶⁴/₃₆₅ × ³⁶³/₃₆₅ × … × ^(366−n)/₃₆₅

People (n)	P(all different)	P(shared birthday)
2	364/365 ≈ 0.9973	≈ 0.27%
10	≈ 0.8831	≈ 11.7%
20	≈ 0.5886	≈ 41.1%
23	≈ 0.4927	≈ 50.7%
30	≈ 0.2937	≈ 70.6%
50	≈ 0.0296	≈ 97.0%
70	≈ 0.00084	≈ 99.9%

📝 Questions

Person 1 enters an empty room. Their birthday can be any of 365 days. What is P(no shared birthday) at this point? 1 mark

Person 2 enters. For no shared birthday, their birthday must avoid the 1 day already taken. Write P(Person 2's birthday is different) as a fraction. 1 mark

Using the multiplication rule, write an expression for P(all 3 people have different birthdays). You do not need to calculate it yet. 2 marks

Write the general formula for P(all n people have different birthdays). Use product notation or expand the first few terms clearly. 3 marks

Calculate P(at least two people share a birthday) for n = 5. Show your working. (Round to 3 decimal places.) 3 marks

Most people guess ~180 when asked how many people are needed for a 50% chance of a shared birthday. The actual answer is 23. Why do you think people's intuition is so far off? 3 marks

Your class has 30 students. Using the probability ≈ 70.6% for n = 30, how many classes out of 100 would you expect to contain at least one shared birthday? 2 marks

Extension: The probability grows quickly because each new person adds many new possible pairs. How many unique pairs of people are there in a group of n people? Write a formula and evaluate it for n = 23. 7 marks

Answer Key

🎂 Birthday Problem

1.P = 365/365 = 1 (certain — no conflict possible with one person).

2.P = 364/365.

3.P = 365/365 × 364/365 × 363/365.

4.P(all different, n people) = ∏ᵢ₌₀ⁿ⁻¹ (365−i)/365 = (365 × 364 × … × (366−n)) / 365ⁿ

5.P(all different, 5) = (365×364×363×362×361)/365⁵ ≈ 0.9729. P(shared) = 1 − 0.9729 ≈ 0.027 = 2.7%.

6.People tend to compare each person to themselves ("does anyone share MY birthday?"), giving odds of 1/365 per person. But the question asks about any pair — with 23 people there are C(23,2) = 253 possible pairs, each with a small chance of matching. The cumulative probability is much higher than expected.

7.70.6% × 100 ≈ 71 classes would have at least one shared birthday.

8.Pairs = C(n, 2) = n(n−1)/2. For n=23: 23×22/2 = 253 pairs. Each pair has a 1/365 ≈ 0.27% chance of matching, and 253 such chances accumulate quickly.

🎂 The Birthday Problem

Answer Key

🎂 Birthday Problem

Answer Key

Birthday Problem