TopCoder problem "BirthdayOdds" used in SRM 174 (Division I Level One , Division II Level Two)

Problem Statement


Here is an interesting factoid: "On the planet Earth, if there are at least 23 people in a room, the chance that two of them have the same birthday is greater than 50%." You would like to come up with more factoids of this form. Given two integers (minOdds and daysInYear), your method should return the fewest number of people (from a planet where there are daysInYear days in each year) needed such that you can be at least minOdds% sure that two of the people have the same birthday. See example 0 for further information.



Parameters:int, int
Method signature:int minPeople(int minOdds, int daysInYear)
(be sure your method is public)


-Two people can have the same birthday without being born in the same year.
-You may assume that the odds of being born on a particular day are (1 / daysInYear).
-You may assume that there are no leap years.


-minOdds will be between 1 and 99, inclusive.
-daysInYear will be between 1 and 10000, inclusive.
-For any number of people N, the odds that two people will have the same birthday (in a room with N people, on a planet with daysInYear days in each year) will not be within 1e-9 of minOdds. (In other words, you don't need to worry about floating-point precision for this problem.)


Returns: 4

We must be 75% sure that at least two of the people in the room have the same birthday. This is equivalent to saying that the odds of everyone having different birthdays is 25% or less.

  1. If there is only one person in the room, the odds are 5/5 or 100% that nobody shares a birthday.
  2. If there are two people in the room, the odds are 5/5 * 4/5 = 80% that nobody shares a birthday. This is because the second person has 4 "safe" days on which his birthday could fall, out of 5 possible days in the year.
  3. If there are three people in the room, the odds of no overlap are 5/5 * 4/5 * 3/5 = 48%.
  4. If there are four people in the room, the odds are 5/5 * 4/5 * 3/5 * 2/5 = 19.2%. This means that you can be (100% - 19.2%) = 80.8% sure that two or more of them do, in fact, have the same birthday.

We only need to be 75% sure of this, which was untrue for three people but true for four. Therefore, your method should return 4.

Returns: 23

The factoid from the problem statement. If there are 22 people in a room, the odds of a shared birthday are roughly 47.57%. With 23 people, these odds jump to 50.73%, which is greater than or equal to the 50% needed.

Returns: 4

Another example from planet Earth. The odds of a repeat birthday among only four people are roughly 1.64%.

Returns: 184

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lbackstrom , brett1479

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