/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 A pair of dice is rolled. The ob... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 13

A pair of dice is rolled. The observation is the number that comes up on each die (see Example 16.4 ). Write out the event described by each of the following statements as a set. (a) \(E_{1}\) " pairs are rolled." (A "pair" is a roll in which both dice come up the same number.) (b) \(E_{2}:\) "craps are rolled." ("Craps" is a roll in which the sum of the two numbers rolled is \(2,3,\) or \(12 .)\) (c) \(E_{3}:\) "a natural is rolled." (A "natural" is a roll in which the sum of the two numbers rolled is 7 or \(11 .\) )

Short Answer

Expert verified
Event \(E_{1}\) = \{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\}\n Event \(E_{2}\) = \{(1,1), (1,2), (2,1), (6,6)\}\n Event \(E_{3}\) = \{(1,6), (6,1), (2,5), (5,2), (3,4), (4,3), (5,6), (6,5)\} for sum of 7 and \{(5,6), (6,5)\} for sum of 11

Step by step solution

01

Define Event \(E_{1}\)

Event \(E_{1}\) is defined as a scenario where pairs are rolled. This means that both dice come up with the same number. Therefore, the possible outcomes that conform to this definition can be represented as: \((1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\)
02

Define Event \(E_{2}\)

Event \(E_{2}\) is defined as a scenario where craps are rolled. This infers that the sum of the numbers on the two dice should be 2, 3, or 12. The outcomes that correspond to this definition are: \((1,1), (1,2), (2,1), (6,6)\)
03

Define Event \(E_{3}\)

Event \(E_{3}\) is described as a scenario where a natural is rolled, meaning the sum of the two numbers on the dice is either 7 or 11. The outcomes conforming to this rule can therefore be represented by the following pairs: \((1,6), (6,1), (2,5), (5,2), (3,4), (4,3), (5,6), (6,5)\) for the sum that equals 7 and \((5,6), (6,5)\) for the sum that equals 11

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Dice Probabilities
Understanding the probabilities of rolling dice can be very engaging. When you roll two dice, each die has six sides, numbered from 1 to 6. Therefore, there are a total of 36 possible outcomes because each die is independent of the other (6 sides on die 1 脳 6 sides on die 2 = 36 outcomes).

Dice probability deals with determining the likelihood of a specific event occurring. For example, rolling a pair is an event where both dice show the same number. Since there are 6 such outcomes, the probability can be calculated using:
  • Probability of rolling a pair = Number of favorable outcomes / Total possible outcomes = 6/36 = 1/6
This helps us determine the chances of events happening when dealing with multiple dice.
Events in Probability
Events are specific outcomes that we are interested in when performing a probability experiment. In the case of dice, these events might include rolling doubles, achieving a certain sum, or something more specific like rolling craps or a natural.

An event can consist of:
  • A single outcome, like rolling a double six (an outcome of (6,6)).
  • Multiple outcomes that fulfill a condition, such as all doubles (pairs), these are \( (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) \).
  • Specific sums of dice, like craps or a natural, requires looking at combinations that total to these sums.
Understanding events helps to define and calculate the probabilities of these scenarios and informs how likely it is that something occurs during a dice roll.
Sum of Dice Outcomes
When rolling two dice, the sum of the numbers is an important concept. Each possible sum has a certain probability depending on how many combinations result in that sum.

For example, to find the sum of 7:
  • The outcomes are: (1,6), (6,1), (2,5), (5,2), (3,4), (4,3).
  • There are 6 outcomes, giving it a higher probability than other sums.
Similarly, other sums like 2, 3, 11, and 12 have different outcomes:
  • Sum of 2: Only (1,1)
  • Sum of 3: (1,2) and (2,1)
  • Sum of 11: (5,6) and (6,5)
  • Sum of 12: Only (6,6)
These sums, their combinations, and probabilities are crucial when understanding games like craps or backgammon, where specific rolls are significant to the outcome.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An urn contains seven red balls and three blue balls. (a) If three balls are selected all at once, what is the probability that two are blue and one is red? (b) If three balls are selected by pulling out a ball, noting its color, and putting it back in the urn before the next selection, what is the probability that only the first and third balls drawn are blue? (c) If three balls are selected one at a time without putting them back in the urn, what is the probability that only the first and third balls drawn are blue?

A coin is tossed three times in a row. The observation is how the coin lands (heads or tails) on each toss [see Exercise \(1(a)]\). Write out the event described by each of the following statements as a set. (a) \(E_{1}\) : "the coin comes up heads exactly twice." (b) \(E_{2}:\) "all three tosses come up the same." (c) \(E_{3}:\) "exactly half of the tosses come up heads." (d) \(E_{4}:\) "the first two tosses come up tails."

A student randomly guesses the answers to a four-question true-or-false quiz. The observation is the student's answer (T or \(F\) ) for each question [see Exercise \(2(\mathrm{~b})\) ]. Write out the event described by each of the following statements as a set. (a) \(E_{1}:\) "the student answers \(T\) to two out of the four questions." (b) \(E_{2}\) : "the student answers \(T\) to at least two out of the four questions." (c) \(E_{3}\) : "the student answers \(T\) to at most two out of the four questions." (d) \(E_{4}:\) "the student answers \(T\) to the first two questions."

The sample spaces are too big to write down in full. In these exercises, you should describe the sample space either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear. A student randomly guesses the answers to a multiplechoice quiz consisting of 10 questions. The observation is the student's answer \((A, B, C, D,\) or \(E)\) for each question. Describe the sample space.

The sample spaces are too big to write down in full. In these exercises, you should describe the sample space either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear. A die is rolled four times in a row. The observation is the number that comes up on each roll. Describe the sample space.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.