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Birthday problem

Solve the first problem 18 problems

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1. Guaranteeing a shared birthday ●○○○○ 10 ptsWhat is the smallest number of people needed in a group to guarantee that two of them share the same birthday? Assume there are 366 possible birthdays, including 29 February.
2. Pigeonhole for shared birthdays ●○○○○ 10 ptsWhat is the smallest number of people needed to be certain that at least two of them have the same birthday? Assume there are 366 possible birthdays (including February 29).
3. Birthday not in December ●○○○○ 10 ptsIgnoring leap years, what is the probability that a randomly chosen person's birthday is not in December? Give your answer as a decimal rounded to three decimal places.
4. Birthday on a given day ●●○○○ 20 ptsA class has 30 students, and each student's birthday is equally likely to be any of 365 days. Find the probability that at least one student has a birthday on 5 November. Round to three decimal places.
5. Guaranteed shared birthday count ●●○○○ 20 ptsAmong 10,000 people, what is the largest number of people who are guaranteed to share the same birthday (ignoring the year)? Count February 29 as a possible birthday; the answer is the same with or without it.
6. Birthday match among three people ●●○○○ 20 ptsThree people are in a room. Assuming each birthday is equally likely among 365 days (ignore leap years), find the probability that at least two of them share the same birthday. Round your answer to 9 decimal places.
7. Smallest group with a shared birthday ●●●○○ 50 ptsWhat is the smallest number of people in a group such that the probability that at least two of them share the same birthday is greater than 50%? Assume there are 365 equally likely birthdays and ignore leap years.
8. Every month has a birthday ●●●○○ 50 ptsAssume 40 people are independent and each birthday is equally likely to fall in any of the 12 months. What is the probability that every one of the 12 months contains at least one of the 40 birthdays? Give your answer as a decimal rounded to seven decimal places.
9. Smallest group with matching last four digits ●●●○○ 50 ptsAssume each person's last four digits are equally likely to be any of 0000 to 9999, independently. What is the smallest number of people needed so that the probability that at least two of them have the same last four digits is at least 99.9%?
10. Expected people until first birthday match ●●●●○ 100 ptsAssume birthdays are independent and uniformly distributed over 365 days. People are questioned one at a time. Find the expected number of people questioned until the first person shares a birthday with someone already questioned (this person is counted). Round to two decimal places.
11. Minimum people covering every birthday ●●●●○ 100 ptsAssume a year has 365 equally likely birthdays, each person's birthday is independent of the others, and leap years are ignored. What is the minimum number of people needed so that the probability that every one of the 365 days is someone's birthday is greater than $0.5$?
12. Birthdays that are equal or adjacent ●●●●○ 100 ptsAssume 23 people have independent birthdays, each equally likely to be any of 365 days (February 29 is not considered), and January 1 and December 31 are considered adjacent. Find the probability that at least two people share the same birthday or have adjacent birthdays. Round to seven decimal places.

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