"The Birthday Problem
Determine the probability that at least 2 people in a room of 10 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 10 people have 10 different birthdays. Hint: The first person's birthday can occur 365 ways, the second person's birthday can occur 364 ways, because he or she cannot have the same birthday as the first person, the third person's birthday can occur 363 ways, because he or she cannot have the same birthday as the first or second person, and so on.
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Verified step by step guidance

1. Understand the problem: We want to find the probability that at least two people in a group of 10 share the same birthday. To do this, it's easier first to find the probability that all 10 people have different birthdays, then use the complement rule.
2. Calculate the total number of possible birthday assignments for 10 people, assuming each person can have any of the 365 birthdays independently. This total number is \$365^{10}$.
3. Calculate the number of favorable outcomes where all 10 people have different birthdays. For the first person, there are 365 possible birthdays, for the second person 364 (since they cannot share the first person's birthday), for the third 363, and so on, down to the tenth person who has \$365 - 9 = 356\( possible birthdays. So, the number of favorable outcomes is \)365 \times 364 \times 363 \times \cdots \times 356$.
4. Express the probability that all 10 people have different birthdays as the ratio of favorable outcomes to total possible outcomes: \(P(\text{all different}) = \frac{365 \times 364 \times 363 \times \cdots \times 356}{365^{10}}\)
5. This probability can also be written using factorial notation as: \(P(\text{all different}) = \frac{\frac{365!}{(365 - 10)!}}{365^{10}}\) This completes the calculation for part (a).

Key Concepts

Complement Rule in Probability

The complement rule states that the probability of an event occurring is one minus the probability that it does not occur. In the Birthday Problem, it's easier to calculate the probability that no two people share a birthday and then subtract this from 1 to find the probability that at least two people share a birthday.

Counting Principles and Permutations

Counting principles help determine the number of ways events can occur. For the Birthday Problem, permutations are used to count the number of ways 10 people can have distinct birthdays by multiplying decreasing available options (365, 364, 363, etc.), reflecting the restriction that no birthdays repeat.

Uniform Probability Distribution

This assumes each birthday is equally likely, with a probability of 1/365 for each day, ignoring leap years. This uniformity allows for straightforward calculation of probabilities by treating each birthday as an independent and equally probable event.