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Chapter 5: Problem 30

Roll a fair six-sided die. a. What is the probability that the die shows an even number or a number less than 4 on top? b. What is the probability the die shows an odd number or a number greater than 4 on top?

Short Answer

Expert verified
a. The probability that the die shows an even number or a number less than 4 on top is \(\frac{5}{6}\) .\n b. The probability the die shows an odd number or a number greater than 4 on top is \(\frac{2}{3}\)

Step by step solution

01

Identify the Total and Successful Outcomes for First Question

The total outcomes when a six-sided die is rolled are six [1,2,3,4,5,6]. The successful outcomes for the first question are those when the number on the die is even or less than 4. The even numbers are [2,4,6] and the numbers less than 4 are [1,2,3]. After combining and removing duplicates, the successful outcomes are [1,2,3,4,6]. Therefore, the total number of successful outcomes are 5.
02

Calculate the Probability for First Question

Probability is calculated by dividing the number of successful outcomes by the number of total outcomes. \(\frac{Number of successful outcomes}{Total number of outcomes} = \frac{5}{6} \) for part a.
03

Identify the Total and Successful Outcomes for Second Question

The total outcomes are the same, [1,2,3,4,5,6]. The successful outcomes for the second question are those when the number on the die is odd or greater than 4. The odd numbers are [1,3,5] and the numbers greater than 4 are [5,6]. After combining and removing duplicates, the successful outcomes are [1,3,5,6]. Therefore, the total number of successful outcomes are 4.
04

Calculate the Probability for Second Question

Probability is calculated by dividing the number of successful outcomes by the number of total outcomes. \(\frac{Number of successful outcomes}{Total number of outcomes} = \frac{4}{6} \) for part b.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dice
A six-sided die is a classic tool used in probability exercises and games around the world. Each side features one of the numbers from 1 to 6. When a die is rolled, each number is equally likely to show up on top, making it a **fair** die. This means that the outcomes 1, 2, 3, 4, 5, and 6 each have a probability of \( \frac{1}{6} \).
When calculating probabilities with dice, it鈥檚 important to identify the total possible outcomes. For instance, a six-sided die has 6 total outcomes, which provide the basis for further calculations of probability in any scenario where the die is used.
Even numbers
Even numbers are those that can be divided by 2 without leaving a remainder. On a six-sided die, the even numbers are:
  • 2
  • 4
  • 6
When you are tasked with determining the probability of rolling an even number with a die, you count these even outcomes. In our specific example, these numbers are 2, 4, and 6.
To find the probability of rolling an even number, divide the number of even outcomes (3) by the total number of possibilities (6): \(\frac{3}{6} = \frac{1}{2} \).This means there's a 50% chance of rolling an even number on a standard six-sided die.
Odd numbers
Odd numbers differ from even numbers in that they have a remainder of 1 when divided by 2. When rolling a six-sided die, the odd numbers you might roll include:
  • 1
  • 3
  • 5
If asked to find the probability of rolling an odd number, you first identify these odd outcomes. In our dice scenario, the numbers 1, 3, and 5 are considered. Thus, the probability calculation involves the 3 odd outcomes out of 6 possible results:\(\frac{3}{6} = \frac{1}{2} \).So, there is also a 50% chance of rolling an odd number with a fair six-sided die.
Successful outcomes
In probability exercises, successful outcomes refer to the outcomes that satisfy the conditions of the problem. To solve probability questions, it is essential to determine which results are classified as "successful."
To solve our initial exercise, let's define successful outcomes based on the given conditions:
  • For part (a): Successful outcomes must be even or less than 4. After eliminating duplicates from combining these criteria, the successful outcomes are [1, 2, 3, 4, 6].
  • For part (b): Successful outcomes must be odd or greater than 4. Again, after combining, the results [1, 3, 5, 6] are counted as successful outcomes.
The probability of an event is calculated as the ratio of successful outcomes to total outcomes. Therefore, the probability for each part is obtained by counting which numbers meet the specified conditions out of the total six possible outcomes of a six-sided die.

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Most popular questions from this chapter

Suppose a student is selected at random from a large college population. a. Label each pair of events as mutually exclusive or not mutually exclusive. i. The students is a Chemistry major; the student works on campus. ii. The student is a full-time student; the student is only taking one 3-unit class. b. Give an example of two events that are not mutually exclusive when a student is selected at random from a large college population.

Suppose a person is selected at random from a large population. a. Label each pair of events as mutually exclusive or not mutually exclusive. i. The person has traveled to Mexico; the person has traveled to Canada. ii. The person is single; the person is married. b. Give an example of two events that are mutually exclusive when a person is selected at random from a large population.

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In 2017 the Pew Research Center asked young adults aged 18 to 29 about their media habits. When asked, "What is the primary way you watch television?" \(61 \%\) said online streaming service, \(31 \%\) said cable/satellite subscription, and 5\% said digital antenna. Suppose the Pew Research Center polled another sample of 2500 young adults from this age group and the percentages were the same as those in 2017 . a. How many would say online streaming services? b. How many would say cable/satellite subscription? c. How many would say cable/satellite subseription or digital antenna? d. Are the responses "online streaming service," "cable/satellite subscription," and "digital antenna" mutually exclusive? Why or why not?
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