Math Problem Statement

We have 13 people in our math class. Find the probability that at least one person has the same birthday as someone else. SHOW ALL WORK TO

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:

  1. Total possibilities for birthdays: There are 365 possible birthdays (ignoring leap years).
  2. 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: 365×364×363××(36512)365 \times 364 \times 363 \times \cdots \times (365 - 12) This can be written as: 365!(36513)!\frac{365!}{(365-13)!}

  1. 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: 36513365^{13}

  2. Probability that no one shares a birthday: Probability (no shared birthday)=365!(36513)!36513\text{Probability (no shared birthday)} = \frac{\frac{365!}{(365-13)!}}{365^{13}}

  3. Probability that at least two people share a birthday: P(at least one shared birthday)=1P(no shared birthday)P(\text{at least one shared birthday}) = 1 - P(\text{no shared 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:

  1. How would the probability change if there were 23 people in the class instead of 13?
  2. What is the logic behind using the complement rule to solve this problem?
  3. How does the number of people in the group affect the probability of a shared birthday?
  4. Can this problem be generalized to other events beyond birthdays?
  5. 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