Problem 77 - Basic Statistics for Business and Economics with Student

Question

A puzzle in the newspaper presents a matching problem. The names of 10 U.S. presidents are listed in one column, and their vice presidents are listed in random order in the second column. The puzzle asks the reader to match each president with his vice president. If you make the matches randomly, how many matches are possible? What is the probability all 10 of your matches are correct?

Step-by-Step Solution

Answer

There are 3,628,800 matches possible; probability of all correct is \(\frac{1}{3,628,800}\).

Step 1: Understand the Matching Problem

You are given 10 U.S. presidents and 10 vice presidents in a random order. You need to determine the total number of possible ways to match presidents to vice presidents.

Step 2: Calculate Total Match Possibilities

For 10 presidents and 10 vice presidents, every president can be paired with any of the 10 vice presidents, creating a total number of match combinations which is given by a permutation. The total number of ways to match them is 10 factorial (\(10!\)).

Step 3: Calculate 10 Factorial

Compute \(10!\) which is the product of all positive integers up to 10: \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800\).

Step 4: Calculate the Probability of Correct Matches

There is only 1 correct way to match all presidents to their correlated vice presidents. Hence, the probability that you make all correct matches when selecting randomly is the ratio of correct outcomes to the total outcomes: \(\frac{1}{10!}\).

Step 5: Compute the Probability Value

Calculate \(\frac{1}{10!}\) using the value we computed for \(10!\). Thus, \(\frac{1}{3,628,800}\), which is approximately 0.000000275 or 2.75 \times 10^{-7}.

Key Concepts

Permutations

In combinatorics, permutations refer to the different ways in which a set of items can be arranged or ordered. Each arrangement or order is called a permutation. When arranging 10 U.S. presidents with their vice presidents, we're looking for all possible arrangements of the pairs.

To find the number of possible permutations, we use a formula called "permutation formula," represented as \( n! \) (n factorial), where \( n \) is the number of items to arrange. For our case of 10 presidents, we calculate \( 10! \), meaning we're arranging 10 distinct presidents and their matching vice presidents.

In general, the concept of permutations helps in understanding problems where order or arrangement is crucial. Examples include seating arrangements, ranking participants in a race, or any scenario where sequence matters.

Probability

Probability deals with the likelihood of an event occurring. In our matching problem, we're interested in the probability of correctly matching all presidents to their respective vice presidents when guesses are made randomly.

This involves finding the ratio of successful outcomes to the total number of possible outcomes. Here, a successful outcome is correctly matching all presidents to their vice presidents once. There are 3,628,800 possible matches, hence the probability is \( \frac{1}{3,628,800} \).

Understanding probability helps in evaluating risks and outcomes in real-world situations. It’s widely applicable across various fields like finance, insurance, science, and everyday decision-making.

Factorial

A factorial, denoted as \( n! \), is the product of an integer and all the integers below it, down to 1. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \). The factorial concept is crucial in determining permutations and combinations.

In our puzzle problem, calculating \( 10! \) gives us the total number of ways to arrange 10 presidents with 10 vice presidents. This equals 3,628,800.

Factorials help simplify the process of counting arrangements and combinations. They're especially vital in statistics and probability, enabling easier calculation of probabilities and assessments of potential scenarios.

Matching Problems

Matching problems in combinatorics involve pairing elements from two sets based on certain criteria. For our puzzle, it's about matching each president with its corresponding vice president.

The challenge is to determine possible ways to make these matches, whether based on a strategic plan or randomly. These problems require logical reasoning as well as computation to derive solutions.

• The number of total possible matchings is represented by permutations.
• We also explore the chance of success, which involves probability calculations.

Matching problems appear in various real-world contexts such as assigning tasks, scheduling, and organizing competitions.