/*! 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 4 Backgammon. Backgammon is a boar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 4

Backgammon. Backgammon is a board game for two players in which the playing pieces are moved according to the roll of two dice. Players win by removing all of their pieces from the board, so it is usually good to roll high numbers. You are playing backgammon with a friend and you roll two 6 s in your first roll and two 6 s in your second roll. Your friend rolls two 3 s in his first roll and again in his second row. Your friend claims that you are cheating, because rolling double 6 s twice in a row is very unlikely. Using probability, show that your rolls were just as likely as his.

Short Answer

Expert verified
Both events have a probability of \(\frac{1}{1296}\), so they are equally likely.

Step by step solution

01

Understanding the Problem

We are tasked with showing that rolling double 6s twice in a row is just as likely as rolling double 3s twice in a row in the game of backgammon. Both scenarios involve consecutive rolls, so we need to calculate the probability for each scenario.
02

Calculating Probability of Double 6s

Firstly, let's calculate the probability of rolling double 6s with two dice. The probability for one die to roll a 6 is \(\frac{1}{6}\). To roll two 6s, the probability is \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\). To roll double 6s twice, the probability is \(\left(\frac{1}{36}\right) \times \left(\frac{1}{36}\right) = \frac{1}{1296}\).
03

Calculating Probability of Double 3s

Now, calculate the probability of rolling double 3s. The probability for one die to roll a 3 is \(\frac{1}{6}\). To roll two 3s, the probability is \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\). To roll double 3s twice, the probability is \(\left(\frac{1}{36}\right) \times \left(\frac{1}{36}\right) = \frac{1}{1296}\).
04

Comparing Probabilities

Both scenarios, rolling double 6s twice and rolling double 3s twice, have the same probability of \(\frac{1}{1296}\). Therefore, neither scenario is more likely than the other.

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 Games
Dice games have been a popular form of entertainment for centuries, offering both strategy and a sprinkle of chance. Backgammon is one such game that relies heavily on the outcome of rolling dice. Understanding the basics can enhance your gameplay and appreciation.
  • Two-Dice Roll Mechanics: In games like Backgammon, players roll two six-sided dice, creating a variety of 36 possible outcomes.
  • Double Rolls: Occasionally, the same number appears on both dice, termed as a "double," which is important for game strategy.
  • Winning Conditions: The objective often includes moving pieces strategically based on dice outcomes and removing them from the board efficiently.
Grasping these fundamentals is pivotal, as dice outcomes dictate moves and can significantly affect your strategy and success rate.
Probability Calculation
Probability is a key concept in dice games, helping players understand the likelihood of different outcomes. Calculating probability involves counting favorable outcomes compared to all possible outcomes.
For example, the probability of rolling a double 6 with two dice can be understood as:
  • Single 6 Probability: The chance of getting a 6 on one die is \(\frac{1}{6}\).
  • Double 6 Probability: The combined probability of rolling two 6s in one go is \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\).
  • Consecutive Rolls: For two consecutive double 6s, multiply their probabilities: \(\frac{1}{36} \times \frac{1}{36} = \frac{1}{1296}\).
This method applies similarly for any double, like double 3s. Understanding these calculations not only improves gameplay but also mathematical intuition.
Statistics Education
Learning probability through dice games offers a practical approach to statistics education. It helps in translating abstract concepts into tangible experiences.

Engagement: Games create an engaging learning environment, making the subject fun and interactive.

Applied Critical Thinking: Calculating probabilities in real-life scenarios, like Backgammon, enhances critical thinking and problem-solving skills.

  • Helps students visualize random variables and outcomes.
  • Encourages competitive learning while understanding the role of chance.
  • Facilitates comprehension of independent events in probability.
These educational benefits illustrate the importance of integrating games into learning, making complex ideas more accessible and enjoyable.

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

\(\mathrm{P}(\mathrm{A})=0.3, \mathrm{P}(\mathrm{B})=0.7\) (a) Can you compute \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})\) if you only know \(\mathrm{P}(\mathrm{A})\) and \(\mathrm{P}(\mathrm{B}) ?\) (b) Assuming that events \(A\) and \(B\) arise from independent random processes, i. what is \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})\) ? ii. what is \(\mathrm{P}(\mathrm{A}\) or \(\mathrm{B}) ?\) iii. what is \(\mathrm{P}(\mathrm{A} \mid \mathrm{B}) ?\) (c) If we are given that \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})=0.1,\) are the random variables giving rise to events \(\mathrm{A}\) and \(\mathrm{B}\) independent? (d) If we are given that \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})=0.1,\) what is \(\mathrm{P}(\mathrm{A} \mid \mathrm{B}) ?\)

Consider the following card game with a well-shuffled deck of cards. If you draw a red card, you win nothing. If you get a spade, you win \(\$ 5\). For any club, you win \(\$ 10\) plus an extra \(\$ 20\) for the ace of clubs. (a) Create a probability model for the amount you win at this game. Also, find the expected winnings for a single game and the standard deviation of the winnings. (b) What is the maximum amount you would be willing to pay to play this game? Explain your reasoning.

Data collected by the Substance Abuse and Mental Health Services Administration (SAMSHA) suggests that \(69.7 \%\) of 18-20 year olds consumed alcoholic beverages in any 58 given year. (a) Suppose a random sample of ten 18-20 year olds is taken. Is the use of the binomial distribution appropriate for calculating the probability that exactly six consumed alcoholic beverages? Explain. (b) Calculate the probability that exactly 6 out of 10 randomly sampled 18- 20 year olds consumed an alcoholic drink. (c) What is the probability that exactly four out of ten \(18-20\) year olds have not consumed an alcoholic beverage? (d) What is the probability that at most 2 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages? (e) What is the probability that at least 1 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages?

The game of roulette involves spinning a wheel with 38 slots: 18 red, 18 black, and 2 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. (a) You watch a roulette wheel spin 3 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin? (b) You watch a roulette wheel spin 300 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin? (c) Are you equally confident of your answers to parts (a) and (b)? Why or why not?

Determine if the statements below are true or false, and explain your reasoning. (a) If a fair coin is tossed many times and the last eight tosses are all heads, then the chance that the next toss will be heads is somewhat less than \(50 \%\). (b) Drawing a face card (jack, queen, or king) and drawing a red card from a full deck of playing cards are mutually exclusive events. (c) Drawing a face card and drawing an ace from a full deck of playing cards are mutually exclusive events.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.