Math Problem Statement
Solution
This problem is known as the "Birthday Problem," where we are asked to find the probability that at least two people in a group of 13 share the same birthday.
To approach this, it’s easier to first calculate the probability that no one shares the same birthday, and then subtract that from 1 to find the probability that at least two people share a birthday.
Steps to solve:
- Total possibilities for birthdays: There are 365 possible birthdays (ignoring leap years).
- No one shares a birthday:
- The first person can have any of the 365 days as a birthday.
- The second person can have any of the 364 days remaining (since they can't share the first person's birthday).
- The third person can have any of the 363 days remaining, and so on.
Thus, for 13 people, the number of ways in which no one shares a birthday is: This can be written as:
-
Total possible birthday combinations: Since each person can have any of 365 birthdays, the total number of possible combinations of birthdays for 13 people is:
-
Probability that no one shares a birthday:
-
Probability that at least two people share a birthday:
Let’s calculate this step-by-step. math
Calculate the probability that no one shares a birthday
total_birthday_combinations = 365**13 no_shared_birthday_combinations = math.factorial(365) / math.factorial(365 - 13)
Probability that no one shares a birthday
prob_no_shared_birthday = no_shared_birthday_combinations / total_birthday_combinations
Probability that at least two people share a birthday
prob_at_least_one_shared_birthday = 1 - prob_no_shared_birthday prob_at_least_one_shared_birthdayThe probability that at least one person in the class of 13 shares a birthday with someone else is approximately 0.1944, or about 19.44%.
If you have any more questions or need further clarification, feel free to ask!
Here are 5 follow-up questions:
- How would the probability change if there were 23 people in the class instead of 13?
- What is the logic behind using the complement rule to solve this problem?
- How does the number of people in the group affect the probability of a shared birthday?
- Can this problem be generalized to other events beyond birthdays?
- How does the birthday paradox relate to probability theory?
Tip: The "birthday paradox" shows how our intuitions about probability can often be off—what seems unlikely (a shared birthday in a small group) is actually more probable than we think!
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Math Problem Analysis
Mathematical Concepts
Probability
Combinatorics
Formulas
P(at least one shared birthday) = 1 - P(no shared birthday)
365! / (365 - 13)!
Theorems
Complement Rule in Probability
Suitable Grade Level
Grades 9-12
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