All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Probability
A happy trio (Posted on 2014-10-09) Difficulty: 3 of 5
What is the probability that on campus of 730 students there exists at least one triplet of students celebrating their birthday on the same date?

Rem: Assume a year of 365 days

No Solution Yet Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution A better approximation | Comment 2 of 3 |
The only way to avoid such a triplet is for each possible birthday to be represented exactly twice. Any birthday which was represented only once would leave 364 birthdays for the remaining 2*364+1=729 students. The extra (+1) student would have to match one of the 364 pairs.

So the sought probability is the complement of the probability (very small) that each birthday is represented by exactly 2 students.

The probability that Jan. 1 is represented by exactly 2 students is:
  (1/365)^2 * (364/365)^728 * C(730,2)
Given the above, the probability that Jan. 2 is represented by exactly 2 students is:
  (1/364)^2 * (363/364)^726 * C(728,2)
etc. through

Dec. 30:
  (1/2)^2 * (1/2)^2 * C(2,2)
After that it's certain Dec. 31 falls into line.

    5   P=1
   10   for I=0 to 363
   20    Fctr=1/(365-I)^2*((364-I)/(365-I))^?(728-2*I)*combi(730-2*I,2)
   30    P=P*Fctr
   40   next
   50   print P
evaluates this but finds the result is essentially zero, making the sought answer the complement of this, 1.  To be more exact, it comes to

44294693483441139112975532973, approximately.

(The probability that a given birthday--such as the Jan. 1 mentioned above-- occurs exactly twice is about 27%. The conditional probabilities rise, as more birthdays are found to be in pairs, rising to 3/8 at the end--even split between Dec 30 and 31, above.)

  Posted by Charlie on 2014-10-09 13:48:49
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (5)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information