Textbook Question
The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = n + 5
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 51
In Exercises 49–52, a single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.
Verified step by step guidance1
Identify the sample space for each roll of the die. Since a single die has 6 faces, each roll has 6 possible outcomes: 1, 2, 3, 4, 5, and 6.
Determine the favorable outcomes for the first roll: rolling an even number. The even numbers on a die are 2, 4, and 6, so there are 3 favorable outcomes.
Determine the favorable outcomes for the second roll: rolling a number greater than 2. The numbers greater than 2 are 3, 4, 5, and 6, so there are 4 favorable outcomes.
Calculate the probability of each event separately. The probability of rolling an even number first is \(\frac{3}{6}\), and the probability of rolling a number greater than 2 second is \(\frac{4}{6}\).
Since the two rolls are independent events, multiply the probabilities of each event to find the combined probability: \(\frac{3}{6} \times \frac{4}{6}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability of Independent Events
When two events occur independently, the probability of both happening is the product of their individual probabilities. Rolling a die twice are independent events, so multiply the probability of the first event by the probability of the second.
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Probability of Multiple Independent Events
Sample Space of a Die Roll
A single roll of a fair six-sided die has six equally likely outcomes: 1 through 6. Understanding this sample space helps determine the probability of specific events, such as rolling an even number or a number greater than 2.
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Event Definition and Counting Favorable Outcomes
To find the probability of an event, identify all outcomes that satisfy the event's condition. For example, even numbers on a die are 2, 4, and 6; numbers greater than 2 are 3, 4, 5, and 6. Counting these helps calculate the event's probability.
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Related Practice
Textbook Question
Use the Binomial Theorem to expand each expression and write the result in simplified form. (x1/3 +x-1/3)3
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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Textbook Question
Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)
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Textbook Question
Express each repeating decimal as a fraction in lowest terms. 0.6 (repeating 6)
Textbook Question
Use the formula for nCr to solve Exercises 49–56. Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?