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.
- If there is only one person in the room, the odds are 5/5 or 100% that nobody shares a birthday.
- 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.
- If there are three people in the room, the odds of no overlap are 5/5 * 4/5 * 3/5 = 48%.
- 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.