Problem 26 - A History of Mathematics: An Introduction

Question

In Euler's analysis of the lottery for the case \(k=2\), determine the general formulas for the "fair" prizes \(a\) and \(b\) for matching two numbers and for matching one number, respectively, in terms of \(n\) and \(t\), where \(t\) tokens are drawn out of a total of \(n\).

Step-by-Step Solution

Answer

Question: Determine the general formulas for the "fair" prizes for matching two numbers and for matching one number, respectively, in Euler's analysis of the lottery case when \(k=2\), \(n\) total numbers, and \(t\) drawn tokens. Answer: The general formulas for the "fair" prizes for matching two numbers (a) and for matching one number (b) are: $$a = \frac{\binom{n}{t}}{\binom{t}{2} \binom{n-t}{t-2}}$$ $$b = \frac{\binom{n}{t}}{t \binom{n-t}{t-1}}$$

Step 1: Calculate the total number of possible combinations

Since we have \(n\) total numbers and \(t\) drawn tokens, the total number of possible ways to draw these tokens is given by the binomial coefficient: $$\binom{n}{t} = \frac{n!}{t!(n-t)!}$$ This value represents the total number of possibilities in a lottery, which we will use to calculate the probabilities in the next steps.

Step 2: Calculate the probability of matching two numbers

To match two numbers out of the \(t\) drawn tokens, we need to have both of those numbers among the drawn tokens, and the remaining \((t-2)\) tokens can be any of the remaining \((n-2)\) available numbers. Therefore, the number of ways to match two numbers is: $$\binom{t}{2} \binom{n-t}{t-2} = \frac{t!(t-1)}{2!(t-2)!} \cdot \frac{(n-t)!}{(t-2)!(n-t-(t-2))!}$$ Then, the probability of matching two numbers \(P(\text{match 2})\) is calculated as: $$P(\text{match 2}) = \frac{\binom{t}{2} \binom{n-t}{t-2}}{\binom{n}{t}}$$

Step 3: Calculate the probability of matching one number

To match one number out of the \(t\) drawn tokens, we need to have that number among the drawn tokens and the remaining \((t-1)\) tokens can be any of the remaining \((n-1)\) available numbers. Therefore, the number of ways to match one number is: $$t \binom{n-t}{t-1} = t \cdot \frac{(n-t)!}{(t-1)!(n-t-(t-1))!}$$ Then, the probability of matching one number \(P(\text{match 1})\) is calculated as: $$P(\text{match 1}) = \frac{t \binom{n-t}{t-1}}{\binom{n}{t}}$$

Step 4: Determine the general formulas for "fair" prizes

The fair prize for matching two numbers \(a\) and for matching one number \(b\) can be calculated using the probabilities of matching in the lottery. To make the lottery fair, we need to multiply the probability of each event by its respective payout. Summing up these expected values should be equal to the total price paid for each chance. Let's assume the price for a chance is 1 unit, so: $$a \cdot P(\text{match 2}) + b \cdot P(\text{match 1}) = 1$$ Substitute the calculated probabilities from steps 2 and 3: $$a \cdot \frac{\binom{t}{2} \binom{n-t}{t-2}}{\binom{n}{t}} + b \cdot \frac{t \binom{n-t}{t-1}}{\binom{n}{t}} = 1$$ Now we have a system of equations for a and b: $$a = \frac{\binom{n}{t}}{\binom{t}{2} \binom{n-t}{t-2}}$$ $$b = \frac{\binom{n}{t}}{t \binom{n-t}{t-1}}$$ These are the general formulas for the "fair" prizes \(a\) and \(b\) for matching two numbers and for matching one number, respectively, in Euler's analysis of the lottery with \(k=2\), \(n\) total numbers, and \(t\) drawn tokens.

Key Concepts

Combinatorics

Combinatorics is a field of mathematics that focuses on counting and arranging elements in sets. It plays a crucial role in problems like lotteries, where you need to consider various combinations of numbers. In the concept of the lottery analyzed by Euler, combinatorics assists in determining how many ways the numbers can be drawn.

• The first step in solving such problems typically involves calculating all possible combinations of drawing tokens from a pool.
• This is computed using the binomial coefficient, denoted as \(\binom{n}{t}\), which represents the number of ways to choose \(t\) items from \(n\) without considering the order.

Once you know the total number of combinations, you can then explore the probabilities of specific outcomes, like matching certain numbers, by using further combinatorial calculations. These probabilities are key to determining whether a game is fair or not as they relate to the prospects of winning and expected returns.

Binomial Coefficient

The binomial coefficient is a fundamental tool in combinatorics, providing a way to calculate how many combinations of a certain number of items can be selected from a larger set. It is often expressed using the notation \(\binom{n}{k}\), where \(n\) is the total number of items and \(k\) is the number of items to choose. Formulaically, it is written as: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(n!\) ("n factorial") is the product of all positive integers up to \(n\). This coefficient appears frequently in probability and statistics problems, particularly when dealing with scenarios where order doesn't matter. In lottery analysis and any problem involving tokens or numbers, they help assess how many possible selections can occur.

• In the original exercise, \(\binom{n}{t}\) gives the total number of ways to choose \(t\) tokens from \(n\).
• The steps further break down into probabilities for matching numbers, using \(\binom{t}{2}\) and \(t\) itself for the combinations to evaluate specific scenarios (e.g., matching 1 or 2 numbers).

Mastering the binomial coefficient is essential for analyzing games that involve random selection, as it underpins your ability to calculate fair outcomes.

Fair Games

The concept of fair games involves ensuring that participants have an equal chance of winning, relative to the cost of playing. In a fair game, the expected cost to the player equals the expected return. This is achieved by setting the prizes such that the average payout is equal to the ticket price.
This idea is fundamental to Euler's lottery analysis, where the goal is to identify fair prizes for matching numbers. This involves complex calculations of probabilities and potential payouts.

• Fairness is calculated by equating the total expected payout to the total cost of tickets. For instance, if a ticket costs 1 unit, then \(a \cdot P(\text{match 2}) + b \cdot P(\text{match 1}) = 1\), where \(a\) and \(b\) are the prizes.
• This ensures that over many games, neither the player nor the house has an inherent advantage.

Understanding fair games is crucial for anyone interested in game theory, economics, or probability, as it underlies how games of chance are structured and how players should evaluate their chances of success.

Lottery Analysis

Analyzing a lottery involves calculating the likelihood of different outcomes and determining fair reward structures. This requires a deep understanding of combinatorial mathematics and probability. Euler's approach simplifies this analysis by establishing formulas for rewards based on how many numbers a player matches. In the original exercise, the analysis begins with formulating probabilities for matching either one or two numbers, relative to the total number of tokens. Using these probabilities allows calculating fair prizes for winners.

• The probabilities help identify how often each matching scenario happens.
• This information is then used to determine what the payout should be to maintain fairness.

Lottery analysis not only involves predicting probabilities but also being able to adapt these predictions to dynamic environments, such as those with varying ticket costs or prize amounts. By understanding these factors, you can analyze how changes in the game's structure influence player decisions and the overall fairness of the game.