What is the birthday paradox?
In a group of just 23 people, the chance that two share a birthday is about 51%. It feels surprising because the number of pairs grows fast: 23 people form 253 pairs, each a chance for a match.
Explain it simply
Most people guess you would need around 180 people. But you are not asking whether someone shares your birthday; you are asking whether any two people match. With 23 people there are 23 x 22 / 2 = 253 pairs.
Again, use the complement: find the chance that all birthdays are different, then subtract from 1.
P(all different) = 365/365 x 364/365 x 363/365 x ... for 23 people.
Example
p = 1.0
for i in range(23):
p *= (365 - i) / 365
print(round(p, 4)) # 0.4927
print(round(1 - p, 4)) # 0.5073So P(at least one shared birthday) is about 50.7%. This ignores leap years and assumes each day is equally likely.
The same maths explains collisions in hashing: a table with far fewer slots than items still gets collisions surprisingly soon.
What to say in the interview
"The birthday paradox shows how quickly collision probability grows. Instead of counting matches, I compute the probability that all 23 birthdays are distinct, about 0.4927, and take the complement, about 0.507. It matters for hash collisions and random ID generation."
Common mistakes
- Answering as if one specific birthday has to match. The question is about any pair.