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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mfonda's profile pic

Matthew Fonda | eNotes Employee

Posted on

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:
neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

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.

 

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