Classical Birthday many people do you think need to be in

QUESTION:

Classical Birthday Problem How many people do you think need to be in a room so that 2 people will have the same birthday (month and day)? You might think it is 366. This would, of course, guarantee it (excluding leap year), but how many people would need to be in a room

so that there would be a 90% probability that 2 people would be born on the same day? What about a 50% probability?

Actually, the number is much smaller than you may think. For example, if you have 50 people in a room, the probability that 2 people will have the same birthday is 97%. If you have 23 people in a room, there is a 50% probability that 2 people were born on the same day!

The problem can be solved by using the probability rules. It must be assumed that all birthdays are equally likely, but this assumption will have little effect on the answers. The way to find the answer is by using the complementary event rule as P(2 people having the same birthday) = 1 - P(all have different birthdays).

For example, suppose there were 3 people in the room. The probability that each had a different birthday would be

\(\frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365}=\frac{365 P_{3}}{365^{3}}=0.992\)

Hence, the probability that at least 2 of the 3 people will have the same birthday will be

\(1-0.992=0.008\)

Hence, for \(k\) people, the formula is

P(at least 2 people have the same birthday)

\(=1-\frac{365 P_{k}}{365^{k}}\)

Using your calculator, complete the table and verify that for at least a 50% chance of 2 people having the same birthday, 23 or more people will be needed.