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Chapter 6: Problem 9

Two six-sided dice, one red and one white, will be rolled. List the possible values for each of the following random variables. a. \(x=\) sum of the two numbers showing b. \(y=\) difference between the number on the red die and the number on the white die (red - white) c. \(w=\) largest number showing

Short Answer

Expert verified
The possible values for the variables are: \(x= [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]\), \(y= [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]\), and \(w= [1, 2, 3, 4, 5, 6]\).

Step by step solution

01

Determining Possible Values for Sum of Two Numbers

Each die has six sides with numbers ranging from 1 to 6, so they can result in sums ranging from \(1+1=2\) to \(6+6=12\). The possible values of \(x\) going from 2 to 12 can be listed in ascending order.
02

Possible Values for Difference Between the Numbers on the Red Die and the White Die

The difference between two dice, considering one is red and the other white, ranges from -5 to 5. This is because the largest negative difference occurs when the red die shows 1 and the white die shows 6 (\(1-6=-5\)), and the largest positive difference occurs when the red die shows 6 and the white die shows 1 (\(6-1=5\)). Hence, the values of \(y\) that must be listed ranges from -5 to 5.
03

Possible Values for The Largest Number Showing

The largest number showing on either die can range from 1 to 6. Even if one die shows a 1, and the other a 6, the largest number is still 6. Hence, the values of \(w\) that one needs to list are 1, 2, 3, 4, 5, 6, in ascending order.

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Key Concepts

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

Probability Distributions
Probability distributions describe how probabilities are distributed over the range of possible outcomes for a random variable. They are crucial in understanding how outcomes occur and the likelihood of each outcome.
In the context of rolling two dice, we consider different random variables. For instance, the sum of the numbers on the dice is a random variable, denoted as \(x\).
  • Each sum value (e.g., 2 through 12) has a certain probability depending on how many combinations can result in that sum.
  • For example, there is only one way to achieve a sum of 2 (1+1), but there are multiple combinations to get a sum of 7 (e.g., 1+6, 2+5, 3+4).
A complete understanding of probability distributions helps in calculating these probabilities effectively and designing predictions about outcomes.
Dice Mathematics
Dice are a common tool for probability exercises because of their simplicity and the fixed number of outcomes. With a six-sided die (singular of dice), each face has an equal chance of appearing when rolled.
When throwing two dice, as in this exercise with one red and one white die, the calculations involve:
  • The sum of the two numbers shown, which varies from 2 to 12 based on the pips (dots) on the faces.
  • The difference between the numbers on the dice, which can shift from -5 to 5, inclusive.
  • The largest number shown, which is simply the highest individual die value in any roll.
Understanding these aspects lays a foundation for mastering more advanced probabilistic models and computations.
Possible Outcomes
Possible outcomes are all likely results that can occur from a random event, such as a dice roll. Identifying possible outcomes forms the base for determining the probability of any specific event.
With two dice:
  • There are 36 possible combinations, as each die has 6 sides and 6 x 6 results in 36 pairs.
  • This affects the sum of the numbers, number difference, and the largest number of an outcome.
The practice of listing all possible outcomes for the dice and understanding their frequencies is key for evaluating probability distributions accurately and comprehending statistical concepts.
Mathematics Education
Mathematics education often introduces probabilities through intuitive exercises like dice rolls. Real-world applications of these exercises help develop critical thinking and statistical reasoning.
Educational practices use such problems:
  • To build an intuition for randomness and chance, crucial for probability and statistics.
  • To encourage logical reasoning by exploring outcomes and their likelihoods.
Incorporating concrete examples from everyday life, such as dice rolling, makes abstract mathematical concepts more accessible and engaging for students, fostering a deeper understanding of essential mathematical principles.

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Most popular questions from this chapter

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You are to take a multiple-choice exam consisting of 100 questions with five possible responses to each question. Suppose that you have not studied and so must guess (randomly select one of the five answers) on each question. Let \(x\) represent the number of correct responses on the test. a. What kind of probability distribution does \(x\) have? b. What is your expected score on the exam? (Hint: Your expected score is the mean value of the \(x\) distribution.) c. Calculate the variance and standard deviation of \(x\). d. Based on your answers to Parts (b) and (c), is it likely that you would score over 50 on this exam? Explain the reasoning behind your answer.
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