# A class contains 33 students. What is the probability (in a percentage) that two of these students have the same birthday? Please help me show work!

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### 2 Answers

This problem is know as the *Birthday Problem* or *Birthday Paradox.*

Note: the above answer answers a slightly different question--it answers the probability that anyone has the same birthday as one specific person--it does not answer what the question asks--the probability of *any* two people sharing the same birthday.

The correct probability here is given by

`P(n) = 1 - (365!) / ((365^n)(365-n)!)`

`P(33) = 1 - (365!) / ((365^33)(365-33)!) ~~77%`

(See link below for derivation)

**Sources:**

29 days are there in a leap year which comes once in 4 years or 365*4+1 = 1461 days.

So the probability of two 2 persons' date of birth is 29th Feb is 1/1461^2. Out of 32 persons there are 32C2 = 32*31/2 = 496 such possibilities. So the required probability is 496/1461^2 *100% = **0.023%** approximately.

The probability that any 2 persons' date of birth is same (but other than 29th Feb) is 1/365^2. There are 32C2 such possibilities. So the required probability is 496/365^2*100% = **0.037%** approximately.