Ch. 4 - Probability

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 4 - Probability

Problem 4.4.23b

Chapter 4, Problem 4.4.23b

Corporate Officers and Committees The Self Driving Unicycle Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO), and chief human resources officer (CHR). It must also appoint a strategic planning committee with five different members. There are 15 qualified candidates, and officers can also serve on the committee.

b. How many different ways can a committee of five be appointed?

Verified step by step guidance

Step 1: Recognize that the problem involves selecting a committee of 5 members from a pool of 15 candidates. Since the order of selection does not matter, this is a combination problem.

Step 2: Recall the formula for combinations, which is given by:

C (n, r) = \frac{n!}{r! (n - r)!}

, where

n

is the total number of candidates and

r

is the number of members to be selected.

Step 3: Substitute the values into the formula. Here,

n = 15

and

r = 5

. The formula becomes:

C (15, 5) = \frac{15!}{5! (15 - 5)!}

Step 4: Simplify the factorials. Compute

15!

5!

, and

10!

. Then simplify the fraction by canceling out common terms in the numerator and denominator.

Step 5: Perform the division to find the total number of combinations. This will give the total number of ways to select a committee of 5 members from 15 candidates.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. In this scenario, we need to choose 5 members from 15 candidates, which is a classic example of a combination problem. The formula for combinations is given by C(n, k) = n! / (k!(n-k)!), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial.

Factorial

Factorial is a mathematical operation that multiplies a number by all positive integers less than it. It is denoted by n! and is essential in calculating combinations and permutations. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is crucial for solving problems involving arrangements and selections in statistics.

Counting Principles

Counting principles are fundamental rules used to determine the number of ways to arrange or select items. In this context, the combination formula is applied to count the different ways to form a committee from a pool of candidates. Mastery of counting principles allows for efficient problem-solving in various statistical scenarios, especially when dealing with large sets of data.

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Fundamental Counting Principle

Related Practice

Textbook Question

Denomination Effect

In Exercises 13–16, use the data in the following table. In an experiment to study the effects of using four quarters versus a \$1 bill, some college students were given four quarters and others were given a \$1 bill, and they could either keep the money or spend it on gum. The results are summarized in the table (based on data from “The Denomination Effect,” by Priya Raghubir and Joydeep Srivastava, Journal of Consumer Research, Vol. 36).

Denomination Effect

b. Find the probability of randomly selecting a student who kept the money, given that the student was given four quarters.

Textbook Question

Births in Vietnam In Vietnam, the probability of a baby being a boy is 0.526 (based on the data available at this writing). For a family having four children, find the following.

b. The probability that all four children are girls.

Textbook Question

In Exercises 21-28, find the probability and answer the questions.

Guessing Birthdays On their first date, Kelly asks Mike to guess the date of her birth, not including the year.

b. Would it be unlikely for him to guess correctly on his first try?

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Textbook Question

Births in the United States In the United States, the true probability of a baby being a boy is 0.512 (based on the data available at this writing). For a family having three children, find the following.

b. The probability that all three children are boys.

Textbook Question

Dice and Coins

c. Find the probability that when a six-sided die is rolled, the outcome is 7.

Textbook Question

Kentucky Derby Odds When the horse Justify won the 144th Kentucky Derby, a \$2 bet on a Justify win resulted in a winning ticket worth \(7.80.

c. If the payoff odds were the actual odds found in part (c), what would be the worth of a \)2 win ticket after the Justify win?

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