/*! 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 17 In Exercises 15-20, find the con... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 17

In Exercises 15-20, find the conditional probabilities of the indicated events when two fair dice (one red and one green) are rolled. The red one is 5 , given that the sum is 6 .

Short Answer

Expert verified
The probability that the red die has a score of 5, given that the sum of the two dice is 6, is \(\frac{1}{5}\) or 0.2.

Step by step solution

01

Determine the sample space

The sample space, denoted by S, includes all possible outcomes when we roll a red die and a green die. Since each die has 6 sides, the sample space has 6 x 6 = 36 possible outcomes, corresponding to the ordered pairs (red die score, green die score).
02

Determine the favorable outcomes for the sum being 6

Now, let's determine the outcomes for the event that the sum of the numbers on both dice being 6. We need to find the pairs (r, g) for which r + g = 6 where r is the red die score and g is the green die score. There are 5 such pairs: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1). Let's denote this event by A.
03

Determine the favorable outcomes for the red die being 5

The event of interest is the red die being 5, given that the sum is 6. To find the favorable outcomes for this event, we look for the pairs in A that have the red die score of 5. Only one pair satisfies this condition: (5, 1). Let's call this event B.
04

Calculate the conditional probability

We want to find the conditional probability P(B|A), which is the probability of event B occurring given that event A has occurred. To calculate this, we use the formula: \[ P(B|A) = \frac{\text{Number of favorable outcomes for event B}}{\text{Number of favorable outcomes for event A}} \] In this case, there is only 1 favorable outcome for event B (red die score is 5), and there are 5 favorable outcomes for event A (sum of the dice is 6). So, the conditional probability is: \[ P(B|A) = \frac{1}{5} \] Thus, the probability that the red die has a score of 5, given that the sum of the two dice is 6, is \(\frac{1}{5}\) or 0.2.

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.

Understanding Sample Space
The term 'sample space' is foundational to understanding probability, and it refers to the set of all possible outcomes in a random experiment. When it comes to rolling two dice, each die has 6 faces, hence the combined sample space consists of 36 possible ordered pairs that represent the outcome of the red and green dice. Each ordered pair is denoted by (red die score, green die score).

For example, the sample space when rolling two dice includes outcomes like (1,1), (1,2), through to (6,6). This space is essential as it represents the universe of all outcomes for which we calculate probabilities. When visualizing, imagine a grid where one axis represents the outcomes of the red die and the other the green die; each cell in the grid is a unique outcome in our sample space.
Identifying Favorable Outcomes
Favorable outcomes are specific outcomes from the sample space that correspond to the event of interest. When we roll two dice and are interested in the event where the sum is 6, we filter the sample space for outcomes that meet this condition. In this case, there are 5 such pairs: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1). These are considered favorable because they satisfy the event condition we're focusing on.

It's important to recognize that the number of favorable outcomes directly impacts probability; the more ways an event can occur, the higher the probability. Teaching students to accurately count these outcomes is crucial as they form the basis for calculating probabilities.
Applying the Conditional Probability Formula
Now, the conditional probability formula comes into play when we鈥檙e dealing with events that are dependent on each other鈥檚 occurrence. In symbolic terms, the conditional probability of an event B, given that another event A has already occurred, is denoted as \( P(B|A) \) and is calculated as:

  • \( P(B|A) = \frac{\text{Number of favorable outcomes for event B}}{\text{Number of favorable outcomes for event A}} \)
In our dice example, the likelihood of the red die showing a 5 given that the sum of the dice is 6 (event A occurred) would use this formula. With only 1 outcome where the red die is 5 within the favorable outcomes of event A, the conditional probability is \( \frac{1}{5} \), illustrating a direct and practical application of the formula. Understanding this concept allows students to navigate more complex problems where events influence each other.

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

Employment You have worked for the Department of Administrative Affairs (DAA) for 27 years, and you still have little or no idea exactly what your job entails. To make your life a little more interesting, you have decided on the following course of action. Every Friday afternoon, you will use your desktop computer to generate a random digit from 0 to 9 (inclusive). If the digit is a zero, you will immediately quit your job, never to return. Otherwise, you will return to work the following Monday. a. Use the states (1) employed by the DAA and (2) not employed by the DAA to set up a transition probability matrix \(P\) with decimal entries, and calculate \(P^{2}\) and \(P^{3}\). b. What is the probability that you will still be employed by the DAA after each of the next three weeks? c. What are your long-term prospects for employment at the DAA? HIIIT [See Example 5.]

Explain how the property \(P\left(A^{\prime}\right)=1-P(A)\) follows directly from the properties of a probability distribution.

Name three pairs of independent events when a pair of distinguishable and fair dice is rolled and the numbers that face up are observed.

A dodecahedral die has 12 faces, numbered \(1-12\). If the die is weighted in such a way that 2 is twice as likely to land facing up as 1,3 is three times as likely to land facing up as 1 , and so on, what is the probability distribution for the face landing up?

If \(80 \%\) of candidates for the soccer team pass the fitness test, and only \(20 \%\) of all athletes are soccer team candidates who pass the test, what percentage of the athletes are candidates for the soccer team?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.