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Chapter 6: Problem 25

Assuming that all years have 365 days and all birthdays occur with equal probability, how large must \(n\) be so that in any randomly chosen group of \(n\) people, the probability that two or more have the same birthday is at least \(1 / 2\) ? (This is called the birthday problem. Many people find the answer surprising.)

Short Answer

Expert verified
The minimum number of people (n) needed in a group so that the probability of two or more people sharing the same birthday is at least \( \frac{1}{2} \) is n = 23.

Step by step solution

01

Calculate the probability none of them share the same birthday.

We will start by finding the probability that no one in the group of n people shares a birthday. This is equivalent to calculating the probability of each person having a unique birthday. For the 1st person, they have 365 unique days to choose from, so the probability is: \( \frac{365}{365} \) For the 2nd person, they have 364 unique days (excluding the 1st person's birthday), so the probability is: \( \frac{364}{365} \) Similarly, for the ith person, there will be (365 - i + 1) unique days. We need to multiply the probabilities for each person to find the total probability of no shared birthdays: \(P_{no\ shared\ birthdays} = \frac{365}{365} \times \frac{364}{365} \times \cdots \times \frac{365-n+1}{365} \) 2.
02

Find the probability of at least two of them sharing the same birthday.

Since we have the probability of no shared birthdays, we can subtract that from 1 to find the probability of at least two people sharing the same birthday: \(P_{shared\ birthdays} = 1 - P_{no\ shared\ birthdays} \) 3.
03

Find the minimum n that satisfies the condition.

Now, we need to find the minimum value of n such that the probability of shared birthdays is greater than or equal to 1/2: \(P_{shared\ birthdays} \geq \frac{1}{2} \) _Substitute the previous equation_: \(1 - P_{no\ shared\ birthdays} \geq \frac{1}{2} \) \(P_{no\ shared\ birthdays} \leq \frac{1}{2} \) By iterating through different values of n, we can find the minimum value of n such that \(P_{no\ shared\ birthdays} \leq \frac{1}{2}\). As we increase the value of n, the chances that no shared birthday will decrease as like having two people with the same birthday will increase. 4.
04

Conclusion.

After calculating and iterating through different values of n, we find that the minimum value of n required to satisfy the condition \(P_{shared\ birthdays} \geq \frac{1}{2}\) is n = 23. Therefore, in a group of at least 23 people, the probability of two or more people sharing the same birthday is at least 1/2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is the branch of mathematics concerned with analyzing random phenomena and determining the likelihood of different outcomes. It is foundational for understanding concepts like the birthday problem, which takes advantage of the idea that events have calculable probabilities based on the outcomes relative to the total number of possible outcomes.

In probability theory, the sum of probabilities for all possible outcomes in a given scenario always equals 1. When solving the birthday problem, we calculate the probability of the complementary event 鈥� that no two people share a birthday 鈥� and then find the probability of at least two people sharing a birthday by subtracting the complementary probability from 1.

Moreover, understanding dependent and independent events is crucial. In the context of the birthday problem, each person's birthday is considered an independent event because one person's birthday does not influence another's. Thus, calculating the joint probability involves multiplying the probabilities of each independent event.
Combinatorics
Combinatorics, a fundamental part of discrete mathematics, is the study of counting, arrangement, and combination of objects. It's essential in solving problems that ask 'How many ways can something occur?' which is a question at the core of the birthday problem.

In this context, combinatorics helps by providing the framework to count all possible unique birthday arrangements for n people. The principle of multiplication is especially relevant; to find the probability for the first person, we consider all 365 days, for the second, 364, and so on. This step-by-step counting falls under the umbrella of combinatorics and is critical to building the foundation needed for calculating probabilities in more complex scenarios.
Discrete Mathematics
Discrete mathematics covers a suite of topics distinct from continuous mathematics, and it involves structures that are fundamentally countable or distinct. The concepts of discrete mathematics, like graph theory, logic, set theory, and combinatorics, come into play in computer algorithms, cryptography, and network design.

In the case of the birthday problem, we are working with a finite number of days (365) and a countable number of people (n), both of which are discrete quantities. We utilize discrete probability distributions to assess the probability of distinct, countable events occurring, like the distribution of birthdays among a group of people. Understanding how to manipulate these discrete variables is key to solving and understanding the birthday problem.

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