Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 90

Chapter 9, Problem 90

A die is rolled. Find the probability of getting a number less than 3 or greater than 4.

Verified step by step guidance

Step 1: Understand the problem. A standard die has six faces numbered 1 through 6. We need to find the probability of rolling a number less than 3 or greater than 4.

Step 2: Identify the numbers that satisfy the condition 'less than 3'. These are the numbers 1 and 2.

Step 3: Identify the numbers that satisfy the condition 'greater than 4'. These are the numbers 5 and 6.

Step 4: Combine the two sets of numbers (less than 3 and greater than 4). The numbers satisfying either condition are 1, 2, 5, and 6.

Step 5: Calculate the probability. Since there are 4 favorable outcomes (1, 2, 5, 6) out of 6 possible outcomes (1 through 6), the probability is the ratio of favorable outcomes to total outcomes: \( \frac{4}{6} \). Simplify the fraction if needed.

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Key Concepts

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

Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this context, it quantifies the chance of rolling a specific outcome on a die. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Sample Space

The sample space is the set of all possible outcomes of a random experiment. For a single roll of a standard six-sided die, the sample space consists of the numbers {1, 2, 3, 4, 5, 6}. Understanding the sample space is crucial for calculating probabilities, as it provides the context for determining favorable outcomes.

Compound Events

Compound events involve the combination of two or more simple events. In this question, we are interested in the event of rolling a number less than 3 (which includes 1 and 2) or greater than 4 (which includes 5 and 6). To find the probability of compound events, we can use the addition rule, ensuring that we account for any overlapping outcomes.

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Complementary Events

Related Practice

Textbook Question

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is a_n = (3)5ⁿ Find a₂/a₃, a₁/a₂, a₄/a₃ and a₅/a₄ What do you observe?

Textbook Question

Evaluate n!/(n-r)! for n = 20 and r = 3

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Textbook Question

Exercises 88–90 will help you prepare for the material covered in the next section. Use the formula an = a₁3^(n-1)to find the seventh term of the sequence 11, 33, 99, 297,...

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Textbook Question

Evaluate n!/(n-r)!r! for n = 8 and r = 3

Textbook Question

You are dealt one card from a 52-card deck. Find the probability of getting an ace or a king.

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Textbook Question

A die is rolled. Find the probability of getting a number less than 5.

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