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Chapter 5: Problem 77

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?

Short Answer

Expert verified
There are 3,628,800 possible matches. The probability of matching all correctly is \( \frac{1}{10!} \).

Step by step solution

01

Understanding the Matching Concept

To solve this problem, we need to assess the number of ways to match each U.S. president with their vice president correctly. Each president can be paired with any of the 10 vice presidents, so we are arranging 10 pairs.
02

Calculating the Total Number of Permutations

The total number of ways to arrange 10 pairs is the factorial of 10, which is denoted as \(10!\). This is calculated as \(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).
03

Evaluating \(10!\)

Calculating \(10!\) gives us the total number of permutations: \(10! = 3,628,800\). This represents the total number of possible matches.
04

Calculating the Probability of a Perfect Match

The probability that all 10 presidents are correctly matched with their vice presidents involves only 1 correct arrangement out of the total possible arrangements. Therefore, the probability is \( \frac{1}{10!} \).
05

Expressing the Probability Value

\( \frac{1}{10!} \) translates to a probability of approximately \(2.755 \times 10^{-7}\), which is a very small chance.

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Key Concepts

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

Permutations
Permutations are a fundamental concept in probability and combinatorics. They refers to the different ways in which a set of objects can be arranged. When dealing with permutations, order matters. This concept is critical in solving problems where arrangement or sequence is important.
For instance, if we have a set of 3 letters: A, B, and C, permutations help us determine how many unique ways we can arrange these letters. This would include sequences such as ABC, BAC, CAB, as well as others. The number of permutations of a set is found by calculating its factorial, which we'll discuss in detail in the next section.
  • Permutations consider the order of arrangement.
  • Used to determine how objects can be ordered or arranged.
  • Critical in solving problems where sequence matters.
Factorial
Factorials are a key mathematical tool in permutations and are used to calculate the number of ways to arrange a set of items. The factorial of a non-negative integer n is denoted as n! and is the product of all positive integers less than or equal to n. For example, the factorial of 5, written as 5!, is calculated as follows:
5! = 5 脳 4 脳 3 脳 2 脳 1 = 120
Factorials grow very rapidly; even a relatively small number like 10! results in a large number, 3,628,800. This is key in calculating permutations, as it represents the total number of possible sequences or arrangements of a given set.
  • Factorials are the product of a series of descending natural numbers.
  • Denoted by n! (e.g., 10!).
  • Essential for calculating permutations and arrangements.
Matching Problems
Matching problems involve pairing sets of elements to find combinations that meet certain criteria, such as correctly linking presidents to vice presidents. For this type of problem, each member of one group must be matched uniquely with a member of another group.
In the given exercise, with 10 presidents, each needs to be matched with one of their 10 vice presidents. The number of possible ways to match each president with a vice president corresponds to the permutations of the 10 pairs, which is a huge number calculated as 10! or 3,628,800 ways.
  • Matching involves unique pairing of elements from two sets.
  • Used to solve problems where each element needs one specific partner.
  • Can be solved by finding permutations for viable pairings.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations, arrangements, and counting. It is the foundation behind understanding permutations, combinations, and various counting problems. Combinatorics provides the tools needed to handle problems involving finite structures.
In the context of permutations and matching, combinatorics helps us calculate the number of possible arrangements and also estimate probabilities. For example, when calculating how many ways 10 presidents can be matched with 10 vice presidents, combinatorics is used to arrive at the factorial calculation and understand concepts such as probability of correct matches.
  • Deals with counting, arrangement, and combination of elements.
  • Helps in evaluating probabilities in matching and arrangement problems.
  • Fundamental in understanding permutations, combinations, and factorials.

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Most popular questions from this chapter

Althoff and Roll, an investment firm in Augusta, Georgia, advertises extensively in the Augusta Morning Gazette, the newspaper serving the region. The Gazette marketing staff estimates that \(60 \%\) of Althoff and Roll's potential market read the newspaper. It is further estimated that \(85 \%\) of those who read the Gazette remember the Althoff and Roll advertisement. a. What percent of the investment firm's potential market sees and remembers the advertisement? b. What percent of the investment firm's potential market sees, but does not remember, the advertisement?

A telephone number consists of seven digits, the first three representing the exchange. How many different telephone numbers are possible within the 537 exchange?

Winning all three "Triple Crown" races is considered the greatest feat of a pedigree racehorse. After a successful Kentucky Derby, Corn on the Cob is a heavy favorite at 2 to 1 odds to win the Preakness Stakes. a. If he is a 2 to 1 favorite to win the Belmont Stakes as well, what is his probability of winning the Triple Crown? b. What do his chances for the Preakness Stakes have to be in order for him to be "even money" to earn the Triple Crown?

A large company must hire a new president. The board of directors prepares a list of five candidates, all of whom are equally qualified. Two of these candidates are members of a minority group. To avoid bias in the selection of the candidate, the company decides to select the president by lottery. a. What is the probability one of the minority candidates is hired? b. Which concept of probability did you use to make this estimate?

A recent survey reported in Bloomberg Businessweek dealt with the salaries of CEOs at large corporations and whether company shareholders made money or lost money. $$ \begin{array}{|cccc|} \hline & \begin{array}{c} \text { CEO Paid More } \\ \text { Than \$1 Million } \end{array} & \begin{array}{c} \text { CEO Paid Less } \\ \text { Than \$1 Million } \end{array} & \text { Total } \\ \hline \text { Shareholders made money } & 2 & 11 & 13 \\ \text { Shareholders lost money } & \underline{4} & 3 & \frac{7}{20} \\ \hline \text { Total } & 6 & 14 & 20 \\ \hline \end{array} $$ If a company is randomly selected from the list of 20 studied, what is the probability: a. The CEO made more than \(\$ 1\) million? b. The CEO made more than \(\$ 1\) million or the shareholders lost money? c. The CEO made more than \(\$ 1\) million given the shareholders lost money? d. Of selecting two CEOs and finding they both made more than \(\$ 1\) million?
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