Problem 57 - Algebra and Trigonometry

Question

Find the probability that in a group of eight students at least two people have the same birthday.

Step-by-Step Solution

Answer

The probability is approximately 6.856%.

Step 1: Understanding the Problem

We are asked to find the probability that in a group of 8 students, at least two have the same birthday. This is a typical problem involving probability theory and the concept of the 'birthday paradox.'

Step 2: Calculate the Probability of No Shared Birthday

To find the probability of at least two students sharing a birthday, it is easier to first calculate the probability that none of the students share a birthday. Assume there are 365 days in a year (ignoring leap years). The first student can have any birthday, the second student can have any birthday except the first student's, and so on.The probability that the second student does not share a birthday with the first student is \( \frac{364}{365} \). Similarly, for the third student, the probability they do not share a birthday with the first two is \( \frac{363}{365} \), and so on until the eighth student.Thus, the probability that no one shares a birthday is:\[P(\text{no shared birthday}) = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \cdots \times \frac{358}{365}\]

Step 3: Calculate the Multiplicative Probability

Compute the product of these probabilities to find \(P(\text{no shared birthday})\):\[P(\text{no shared birthday}) = 1 \times \frac{364}{365} \times \frac{363}{365} \times \frac{362}{365} \times \frac{361}{365} \times \frac{360}{365} \times \frac{359}{365} \times \frac{358}{365}\]Completing this calculation gives us approx. \(P(\text{no shared birthday}) \approx 0.93144\).

Step 4: Calculate the Probability of at Least One Shared Birthday

The probability that at least two students have the same birthday is the complement of the probability that no students share a birthday. Thus:\[P(\text{at least one shared birthday}) = 1 - P(\text{no shared birthday})\]Substitute the previously found probability:\[P(\text{at least one shared birthday}) = 1 - 0.93144 \approx 0.06856\]

Step 5: Conclusion

Based on the calculations, the probability that in a group of 8 students, at least two share the same birthday is approximately 0.06856, or 6.856%.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with understanding how likely events are to happen. It provides us with the tools to measure the likelihood of different outcomes. In the context of the birthday paradox, which is the probability of at least two people sharing a birthday in a group, probability theory helps us calculate seemingly counterintuitive results. It begins with understanding the total number of possible outcomes (e.g., 365 different birthdays in a year) and assessing the structure of these outcomes relative to the event in question.
The key here is the approach of complement probability, where instead of directly calculating the probability of a group sharing a birthday, we initially find the probability that no one shares a birthday. This is because dealing with complement events can often be more straightforward. In formula form, the probability of sharing a birthday is given by:

• \( P(\text{at least one shared birthday}) = 1 - P(\text{no shared birthday}) \)

Probability theory not only allows us to calculate such probabilities but also paves the way to understanding more complex statistical concepts.

Shared Birthdays

The idea of shared birthdays is central to what is often referred to as the 'birthday paradox.' Despite what one may intuitively think, it only takes a relatively small group for there to be a good chance that at least two people share the same birthday. This is because of the underlying combinatorial probability involved.

• With each additional person in a group, the probability that at least two people have the same birthday increases.
• In this classic problem, one calculates the probability that no two of the 8 people have the same birthday first before finding the desired probability of at least one shared birthday.

Understanding shared birthdays not only challenges preconceived notions of probability but also enriches one's appreciation for probabilistic reasoning in real-world scenarios, such as data collision in hashing algorithms or coding theory. Hence, shared birthdays highlight the non-linear nature of probability growth as a function of group size.

Combinatorial Probability

Combinatorial probability involves counting different ways events can occur and using these counts to determine the probability of those events. It forms the backbone of calculating situations like the shared birthdays problem. To solve for the probability of at least two shared birthdays among 8 students, we must count the ways everyone can have distinct birthdays:

• Each subsequent student has one less day available than the student before.
• This leads to a series of multiplying fractions, each representing the probability of a new student having a unique birthday.

Therefore, the probability that no one shares a birthday among 8 students is a multiplication of these fractional probabilities, specifically \( \frac{365}{365} \times \frac{364}{365} \times \ldots \times \frac{358}{365} \). This results in approximately 0.93144, illustrating the compactness of probabilities with large combinatorial possibilities. Combinatorial probability doesn't just apply to birthday problems but is a cornerstone of all kinds of probabilities where arrangements and selections matter, from card games to logistical planning.