a) Suppose there are 10 people in a room. If you know no one was born on the lea
ID: 3275395 • Letter: A
Question
a) Suppose there are 10 people in a room. If you know no one was born on the leap day (Feb 29), and everyone is equally likely to have any of the other 365 days as their birthday, what's the probability that no two people have the same birthday? b) Suppose now that there are 30 people in the room, with the same restrictions. Again, what's the probability that no two people have the same birthday? c) Now, suppose there are n people in the room. How large does n have to be to ensure that the probability that no two people have the same birthday is smaller than 0.05?Explanation / Answer
There are 365 possible birthdays. (To keep the numbers simpler, we’ll ignore leap years.) The key to assigning the probability is to think in terms of complements: “Two (or more) people share a birthday” is the complement of “All people in the group have different birthdays.” Each probability is 1 minus the other.
The first person could have any birthday (p = 365÷365 = 1), and the second person could then have any of the other 364 birthdays (p = 364÷365).
Multiply those two and you have about 0.9973 as the probability that any two people have different birthdays, or 10.9973 = 0.0027 as the probability that they have the same birthday.
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