What is the Birthday Paradox?

The Birthday Paradox asks: in a group of N randomly chosen people, what's the probability that at least two share the same birthday?

Intuitively, you might think you need 183 people (half of 365) for a 50-50 chance. But the real answer is just 23 people. With 23 people, there's a 50.7% chance of a shared birthday. With 70 people, it's 99.9%. That's the paradox — the number is much smaller than intuition suggests.

The key insight: we're not asking if someone shares your birthday. We're asking if any pair in the group shares a birthday. There are 23 choose 2 = 253 possible pairs, which is a lot of opportunities for a match.

In our puzzle, you compute the probability for a given N, round to an integer percentage, and that percentage mod 26 gives you a letter.

A Concrete Example

Let's work through the classic case: N = 23 people.

Step 1: Compute P(no match)

It's easier to compute the complement — the probability that everyone has a different birthday:

P(no match) = (365/365) × (364/365) × (363/365) × ... × (343/365)

Step by step:
Person 1: 365/365 = 1.0000  (any birthday is fine)
Person 2: 364/365 = 0.9973  (different from person 1)
Person 3: 363/365 = 0.9945  (different from both)
Person 4: 362/365 = 0.9918
...
Person 23: 343/365 = 0.9397

Multiply all together:
P(no match) = 1.0000 × 0.9973 × 0.9945 × ... × 0.9397
            ≈ 0.4927

Step 2: Compute P(match)

P(match) = 1 - 0.4927 = 0.5073 = 50.73%

Step 3: Round to integer percentage and mod 26

51 % 26 = 25 → 'z' (a=0, b=1, ..., z=25)

The answer letter is 'z' (since 51 mod 26 = 25, and 25 = z).

Why It Works: Probability in One Paragraph

Probability multiplies for independent events. The chance that person 2 has a different birthday from person 1 is 364/365. The chance that person 3 has a different birthday from both is 363/365 (two birthdays are "taken"). Multiply these fractions for all N people, and you get the probability that no one shares a birthday. Subtract from 1 to get the probability that at least one pair shares a birthday. The product drops quickly because each new person has fewer "free" birthdays available — with N people, there are N×(N-1)/2 pairs, and each pair has a 1/365 chance of matching.

Solving Tips

• Start with p = 1.0 and loop: p *= (365 - i) / 365 for i in 0..N-1
• This gives P(no match). The answer is 1 - p = P(match)
• Round to integer percentage: round(P(match) × 100)
• Take percentage mod 26 and convert to letter a-z
• For N=23, P ≈ 50.7% → 51 → mod 26 = 25 → z
• For small N (like 10), P is only ~12% → 12 → mod 26 = 12 → m

Difficulty Table

Real-World Applications

• Cryptographic Hash Collisions: The birthday attack on hash functions exploits exactly this math. For a hash with output size m bits, you only need ~2^(m/2) random hashes to find a collision — that's the birthday paradox in action.
• A/B Testing: When running experiments with many variants, the chance of false positives increases dramatically. The birthday paradox explains why multiple comparison corrections are necessary.
• Genetics: The probability that two people share a genetic marker follows the same math. Forensic DNA databases use this to assess random match probabilities.
• Network Security: The birthday paradox explains why 128-bit symmetric keys are considered only 64 bits of security against birthday attacks.
• Lotteries and Gambling: The probability of "close" numbers in lottery draws is much higher than intuition suggests — same math, different domain.

Want to Try It?

Head over to the puzzle browser page. If a Birthday Paradox puzzle is active, the puzzle browser will give you N. Compute the probability iteratively, round to a percentage, and mod to get your letter!