Question

For the birthday problem, Example 1.3.3, use the given \(\mathrm{R}\) function bday to determine the value of \(n\) so that \(p(n) \geq 0.5\) and \(p(n-1)<0.5\), where \(p(n)\) is the probability that at least two people in the room of \(n\) people have the same birthday.

Short answer

The minimum number of people (n) needed in the room so that the probability (p(n)) of at least two people sharing the same birthday is greater than or equal to 0.5 is found by using the 'bday' function and increasing 'n' until p(n) >= 0.5 and p(n-1) < 0.5. The exact value of 'n' may vary depending on the implementation of the 'bday' function.

Step-by-step solution

Initialization

Use the R function 'bday' and initialize the number of people 'n' to 1.

Calculate p(n)

Calculate the probability p(n) that at least two people share the same birthday for n people using the function 'bday(n)'.

Check Probability

Check if the probability p(n) is greater than or equal to 0.5. If it is, continue to Step 4. If it is not, increment n by 1 and repeat Step 2.

Validate the Result

Recalculate p(n-1) to verify that p(n-1) is indeed less than 0.5. This is to ensure that n is the minimum number of individuals for whom the probability of sharing a birthday exceeds 0.5.

Output n

If p(n) >= 0.5 and p(n-1) < 0.5, the number 'n' is the desired result. Output 'n' as the minimum number of people needed.

Key concepts

Mathematical Statistics

Mathematical statistics involves the application of mathematics to statistics, which implies the collection, analysis, interpretation, and presentation of data. A crucial part of this discipline is understanding how to determine the likelihood of events, which can be expressed as probabilities.

One famous illustration of probability in mathematical statistics is the birthday problem, which investigates the probability of two people having the same birthday in a group. Despite seeming counterintuitive, the probability that at least two individuals share the same birthday in a room of 23 people goes surpassing the 50% threshold. This is an excellent example of how statistical analysis can reveal unexpected patterns in seemingly random events.

To determine this probability, statisticians use formulas derived from probability theory. The calculations can become complex, often requiring computational tools to handle the iterative process that is not practical to do by hand. This illustrates the blend of theoretical knowledge and practical computing that characterizes modern statistical analysis.

Probability Theory

Probability theory is the mathematical framework for quantifying uncertainty. It allows us to make predictions about the likelihood of various outcomes. When it comes to the birthday problem, whose formal investigation falls under this theory, we are interested in the probability that in a group of n people, at least two share the same birthday.

The birthday paradox, as it's often called because of its surprisingly small 'n' value, uses the principles of combinatorics and probability to calculate the odds. The probability is computed by considering not just the chances of people not sharing a birthday and subtracting this from certainty (or '1').

To solve the problem, one would use the formula for the probability of at least two people sharing a birthday, which is a complementary of everyone having a unique birthday: \[ p(n) = 1 - \frac{365}{365} \cdot \frac{364}{365} \cdot \ldots \cdot \frac{365-n+1}{365} \]. As 'n' increases, the probability rises rapidly because of the growing number of possible pair combinations that could have the same birthday, illustrating the non-intuitive but deterministic nature of probability theory.

R Programming Language

The R programming language is a powerful tool for statistical computing and graphics, often used by statisticians for data analysis and developing statistical software. In the context of the birthday problem, R's ability to rapidly compute probabilities and manipulate data makes it an ideal choice for iterating through potential outcomes and conducting complex calculations.

An R function like 'bday' mentioned in the original exercise would be designed to quickly calculate the probability that two people in a group of 'n' share the same birthday. This streamlined approach using R demonstrates how computational tools can perform repetitive tasks efficiently — in this case, incrementing 'n' and re-evaluating the probability until specific conditions are met.

By utilizing such functions, students and statisticians alike can focus more on the analysis and interpretation of results, rather than getting bogged down with the minutiae of calculation. Furthermore, R is also known for its extensive libraries of packages that can support a wide variety of statistical techniques, making it an essential language for anyone working in the field of mathematical statistics and beyond.