Textbook Question
Use the formula for nCr to solve Exercises 49–56. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 49
In Exercises 49–52, a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.
Verified step by step guidance1
Understand that the problem involves two independent events: rolling a 2 on the first roll and rolling a 3 on the second roll of a single die.
Recall that the probability of rolling any specific number on a fair six-sided die is \(\frac{1}{6}\).
Calculate the probability of rolling a 2 on the first roll, which is \(\frac{1}{6}\).
Calculate the probability of rolling a 3 on the second roll, which is also \(\frac{1}{6}\).
Since the two rolls are independent, multiply the probabilities of each event to find the combined probability: \(\frac{1}{6} \times \frac{1}{6}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability of a Single Event
Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. For a fair six-sided die, the probability of rolling any specific number, like a 2, is 1/6.
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Probability of Mutually Exclusive Events
Independent Events
Two events are independent if the outcome of one does not affect the outcome of the other. Rolling a die twice involves independent events because the result of the first roll does not influence the second.
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Multiplication Rule for Independent Events
When two events are independent, the probability of both occurring is the product of their individual probabilities. To find the chance of rolling a 2 first and a 3 second, multiply 1/6 by 1/6, resulting in 1/36.
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Related Practice
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