Question
Determine the probability that at least 2 people in a room of 12 people share the same birthday, ignoring leap years and assuming each birthday is equally likely, by answering the following questions:
(a) Compute the probability that 12 people have different birthdays.
(b) The complement of “12 people have different birthdays” is “at least 2 share a birthday”. Use this information to compute the probability that at least 2 people out of 12 share the same birthday.
(a) The probability that 12 people have different birthdays is 28.83 of 34 points
(Round to four decimal places as needed.)
Determine the probability that at least 2 people in a room of 12 people share the same birthday, ignoring leap years and assuming each birthday is equally likely, by answering the following questions:
(a) Compute the probability that 12 people have different birthdays.
(b) The complement of “12 people have different birthdays” is “at least 2 share a birthday”. Use this information to compute the probability that at least 2 people out of 12 share the same birthday.
(a) The probability that 12 people have different birthdays is 28.83 of 34 points
(Round to four decimal places as needed.)
Ask by Mckenzie Garza. in the United States
Mar 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
(a)
The probability that all 12 people have different birthdays is approximately 0.8327.
The probability that all 12 people have different birthdays is approximately 0.8327.
(b)
The probability that at least two people share the same birthday is approximately 0.1673.
The probability that at least two people share the same birthday is approximately 0.1673.
Solution
(a)
To compute the probability that 12 people have different birthdays, we assume there are 365 equally likely birthdays and ignore leap years. The first person can have any birthday. The second person must have a birthday different from the first, so there are 364 choices, the third person has 363 choices, and so on. In general, the probability that all 12 birthdays are different is
To compute the probability that 12 people have different birthdays, we assume there are 365 equally likely birthdays and ignore leap years. The first person can have any birthday. The second person must have a birthday different from the first, so there are 364 choices, the third person has 363 choices, and so on. In general, the probability that all 12 birthdays are different is
This can also be written as the product
Evaluating the product (or using a calculator), we find
(Rounded to four decimal places.)
(b)
The event that at least two people share the same birthday is the complement of the event that all birthdays are different. Therefore, the probability that at least two share the same birthday is
The event that at least two people share the same birthday is the complement of the event that all birthdays are different. Therefore, the probability that at least two share the same birthday is
(Rounded to four decimal places.)
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Extra Insights
To solve the problem, let’s first calculate the probability that 12 people have different birthdays.
(a) There are 365 days in a year, and the first person can have their birthday on any of those days. The second person must have a birthday on a different day, giving them 364 options, and so on. The number of favorable outcomes for 12 people having different birthdays is:
The total number of outcomes when each person can have any birthday (365 days) is:
Thus, the probability
(b) To find the probability that at least 2 people share a birthday, we use the complement of the probability that all 12 have different birthdays:
Thus, the probability that at least 2 out of 12 people share the same birthday is approximately 0.7500 or 75%.
