Problem 9 - An Introduction to Mathematical Statistics and Its Applications

Question

The great English diarist Samuel Pepys asked his friend Sir Isaac Newton the following question: Is it more likely to get at least one 6 when six dice are rolled, at least two 6 's when twelve dice are rolled, or at least three 6 's when eighteen dice are rolled? After considerable correspondence [see (167)], Newton convinced the skeptical Pepys that the first event is the most likely. Compute the three probabilities.

Step-by-Step Solution

Answer

To calculate the probabilities: For six dice, the probability is `1 - (5/6)^6`. For twelve dice, the answer is `1 - C(12, 0) \times (1/6)^0 \times (5/6)^{12} - C(12, 1) \times (1/6)^1 \times (5/6)^{11}`. For eighteen dice, it is `1 - C(18, 0) \times (1/6)^0 \times (5/6)^{18} - C(18, 1) \times (1/6)^1 \times (5/6)^{17} - C(18, 2) \times (1/6)^2 \times (5/6)^{16}`.

Step 1: Understand the binomial distribution

A binomial distribution can help in determining the probability of an event occurring a specified number of times in a fixed number of trials. In this case, the 'trial' is a roll of the dice, and the 'event' is rolling a 6. The probability of rolling a 6 on a fair six-sided dice is \(1/6\). The PMF of a binomial is given by \(P(k; n, p) = C(n, k) \times p^k \times (1 - p)^{n-k}\), where \(C(n, k)\) is the binomial coefficient.

Step 2: Compute the probability for six dice

The event is getting 'at least one 6'. We can find 1 - the probability of 'no 6'. The PMF when \(k=0\) gives the probability of 'no 6'. Here, \(n=6\), \(p=1/6\), and \(k=0\). We calculate as follows: \(1 - C(6, 0) \times (1/6)^0 \times (5/6)^6 = 1 - (5/6)^6\)

Step 3: Compute the probability for twelve dice

The event is 'at least two 6s'. We find 1 - the probability of 'less than two 6s'. This is `1 - P(0; 12, 1/6) - P(1; 12, 1/6)`. Here, \(n=12\), \(p=1/6\), and \(k=0\) and 1. We calculate as follows: \(1 - C(12, 0) \times (1/6)^0 \times (5/6)^{12} - C(12, 1) \times (1/6)^1 \times (5/6)^{11}\)

Step 4: Compute the probability for eighteen dice

The event is 'at least three 6s'. We find 1 - the probability of 'less than three 6s'. This is `1 - P(0; 18, 1/6) - P(1; 18, 1/6) - P(2; 18, 1/6)`. Here, \(n=18\), \(p=1/6\), and \(k=0\), 1 and 2. We calculate as follows: \(1 - C(18, 0) \times (1/6)^0 \times (5/6)^{18} - C(18, 1) \times (1/6)^1 \times (5/6)^{17} - C(18, 2) \times (1/6)^2 \times (5/6)^{16}\)

Key Concepts

Introduction to Probability Theory

Probability theory is the mathematical study of chance and randomness. When you roll a six-sided die, each face has an equal chance of appearing, so the probability of rolling a 6 is \(\frac{1}{6}\). Probability theory helps us calculate the likelihood of events occurring, especially when outcomes are uncertain.

For example, in Samuel Pepys' question, we want to know the probability of rolling at least a certain number of 6's within multiple dice rolls. This kind of problem is commonly approached using probability to determine how likely different outcomes are.

• Event: An occurrence we're interested in (e.g., rolling at least one 6).
• Trial: A single attempt or roll of the dice.
• Outcome: The result after an experiment (e.g., rolling a number other than 6).

Probability enables us to quantify these situations and assess the likelihood of events, a fundamental skill when dealing with chance scenarios like games, risk assessments, or statistical models.

Understanding Combinatorics

Combinatorics is a branch of mathematics focused on counting and arranging. It becomes particularly handy in scenarios like Pepys' question where we need combinations to solve probability problems. Combinatorics gives us tools to calculate possible configurations and sequences.

The binomial coefficient, represented as \(C(n, k)\) or sometimes \(\binom{n}{k}\), is a key combinatorial concept. It tells us how many ways we can pick \(k\) successes (like rolling a 6) from \(n\) trials (dice rolls).
Understanding combinatorics helps us solve these problems effectively by using formulas to compute probabilities. For example, when calculating the probability of rolling at least three 6s with eighteen dice, we use the binomial coefficients to understand all the possible combinations of dice rolls that meet this condition.

• Factorial: Often used in combinatorial calculations, denoted as \(n!\), representing the product of all positive integers up to \(n\).
• Permutation: Arranging objects in a particular order.
• Combination: Selecting objects where order does not matter, like our dice-rolling scenarios.

By mastering combinatorics, we improve our ability to solve complex probability problems.

Role of Statistical Analysis in Probability

When faced with questions involving probability, statistical analysis provides the methods to test and interpret data effectively. It enables us to make informed decisions based on numerical evidence and testing. With Pepys' dice problem, statistical analysis helps us compare different scenarios and determine which is most probable.

Statistical analysis involves calculating and comparing probabilities for specific outcomes, such as rolling at least one, two, or three 6's in varying numbers of dice rolls. By computing these probabilities and comparing them, we can give an evidence-based answer.

• Descriptive Statistics: Summarize data through mean, median, mode etc.
• Inferential Statistics: Make predictions or inferences about a population based on a sample.
• Probability Distribution: A function that describes the likelihood of different outcomes in a sample space.

By using statistical analysis, we can transform raw data into meaningful insights, crucial for making sense of probability and its practical applications.