Problem 20 - An Introduction to Mathematical Statistics and Its Applications

Question

A corporate board contains twelve members. The board decides to create a five- person Committee to Hide Corporation Debt. Suppose four members of the board are accountants. What is the probability that the Committee will contain two accountants and three nonaccountants?

Step-by-Step Solution

Answer

The calculated probability can vary based on the exact values of combinations, the exact result will have to be calculated either using a calculator or manually.

Step 1: Determine the possible combinations for the committee to consist of two accountants

This is given by the combination \(_4C_2\), where 4 is the number of accountants and 2 is the number of accountants required on the committee. The formula for combination is \(C(n, r) = n! / [(n - r)! * r!]\), where 'n' is the total number of items, 'r' is the items we're selecting, and '!' means factorial.

Step 2: Determine the possible combinations for the committee to consist of three non-accountants

This is given by the combination \(_8C_3\), where 8 is the number of non-accountants (12 total members - 4 accountants), and 3 is the number of non-accountants required in the committee. The formula for combination is same as in Step 1.

Step 3: Determine the total number of ways to form a committee of 5 members

This is given by the combination \(_{12}C_5\), where 12 is the total members and 5 is the number of members required in the committee. As before, the formula for combination is applied.

Step 4: Calculate the Probability

The probability that the committee consists of two accountants and three non-accountants is obtained by dividing the product of the combinations obtained in Steps 1 and 2 by the combination obtained in Step 3.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. It is the art of counting, arranging and grouping things in specific orders or sets without actually having to enumerate every possible combination or permutation.

Understanding problems in combinatorics involves various strategies including the fundamental counting principle, arrangements with permutations, and selections with combinations. For example, when we need to form a committee from a larger group, we're dealing with combinations since the order of selection does not matter. The essence of combinatorial problems is to figure out the number of ways to combine objects from a set without considering sequence.

One of the powerful aspects of combinatorics is its application to probability. By counting the number of favorable outcomes and the total number of possibilities, we can determine the likelihood of specific events occurring. The exercise given is a classic example of applying combinatorial principles to solve a real-world problem using probability.

Factorial Notation

Factorial notation is used to represent the product of an integer and all the non-zero integers below it. It's denoted by the symbol '!'. A factorial of a non-negative integer n, written as n!, is the product of all positive integers less than or equal to n.

For instance, 5! (read as 'five factorial') is calculated as: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials are fundamental to combinatorics as they are used in the formulas for permutations and combinations. Understanding factorial notation is critical when dealing with binomial coefficients, where computing the factorial of numbers is a routine process. Due to its heavy usage, factorial calculation is also embedded into most modern calculators and combinatorial software.

The concept of factorial grows rapidly with higher integers—this rapid growth rate underscores why factorials are common in combinatorial expressions, representing large counts of possible arrangements or selections.

Binomial Coefficient

The binomial coefficient is a key concept in combinatorics, represented as C(n, r), and is pronounced as 'n choose r'. It gives us the number of ways to choose r elements from a larger set of n distinct elements, which is essential in calculating combinations.

The formula to calculate a binomial coefficient is: \[C(n, r) = \frac{n!}{(n-r)! \times r!}\]

• Here, \(n!\) is the factorial of the total number of items.
• \((n-r)!\) is the factorial of the difference between the total items and the items chosen.
• And \(r!\) is the factorial of the number of items chosen.

For example, if we want to find the number of ways to choose 2 accountants out of 4 (as in our exercise), we would calculate \(_4C_2\) using the binomial coefficient formula. This concept directly relates to probabilities since it can define the number of successful outcomes over the set of all possible outcomes. In the example provided, the probability calculation essentially involved finding the appropriate binomial coefficients to represent selecting members for the committee.