/*! 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 3 Create a tree diagram with proba... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 3

Create a tree diagram with probabilities showing outcomes when drawing two marbles without replacement from a bag containing one blue and two red marbles. (You do not replace the first marble drawn from the bag before drawing the second.) (a)

Short Answer

Expert verified
The tree diagram has outcomes: Blue-Red: \(\frac{1}{3}\), Red-Blue: \(\frac{1}{3}\), Red-Red: \(\frac{1}{3}\).

Step by step solution

01

Identify the Initial Setup

We start with a bag containing three marbles: one blue and two red. The total number of marbles initially is 3.
02

Determine First Draw Outcomes

On the first draw, we can either draw a blue marble or a red marble. The probability of drawing a blue marble is \(\frac{1}{3}\), and the probability of drawing a red marble is \(\frac{2}{3}\).
03

Calculate Outcomes After Drawing Blue First

If we draw a blue marble first, only two red marbles remain in the bag. Therefore, the probability of drawing a red marble next is \(1\) (or \(\frac{2}{2}\)).
04

Calculate Outcomes After Drawing Red First

If we draw a red marble first, there will be one blue and one red marble left in the bag. The probability of drawing a blue marble next is \(\frac{1}{2}\), and the probability of drawing another red marble is \(\frac{1}{2}\).
05

Construct the Tree Diagram

The tree diagram starts with the first draw having two branches: 'Blue' with probability \(\frac{1}{3}\) and 'Red' with probability \(\frac{2}{3}\). From the 'Blue' branch, draw a single branch for 'Red' with probability \(1\). From the 'Red' branch, draw two branches: one for 'Blue' with probability \(\frac{1}{2}\) and one for 'Red' with probability \(\frac{1}{2}\).
06

Verify Total Probability

Check that the total probability sums to 1. The possible sequences are Blue-Red, Red-Blue, and Red-Red with respective probabilities of \(\frac{1}{3}\), \(\frac{1}{3}\times\frac{1}{2}\), and \(\frac{1}{3}\times\frac{1}{2}\), which add up to 1.

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.

Conditional Probability
Understanding conditional probability is key to grasping outcomes in this exercise. Conditional probability refers to the likelihood of an event occurring given that another event has already happened.
For example, drawing two marbles without replacement involves conditional probability because the result of the first draw affects the second.
This means the probability for the second marble changes based on the first draw's outcome.
  • If a blue marble is drawn first, the bag only contains red marbles thereafter, making the probability of drawing a red marble on the second draw certain ( \(1\)).
  • In contrast, if a red marble is drawn first, the bag retains one blue and one red marble. Consequently, each has an equal chance of being drawn as the second marble, resulting in probabilities of \(\frac{1}{2}\) for both options.
The calculation process modifies the probability based on what happened before, encapsulating the essence of conditional probability.
Combinatorics
In this context, combinatorics helps us identify the variety of outcomes and paths when drawing marbles.
It essentially deals with counting possibilities and combinations in mathematical problems, which is crucial for constructing tree diagrams like this one.
  • Each node on the tree diagram represents a branching point or an outcome possibility.
  • We start with three marbles, and each draw creates subsequent outcomes, calculated based on how many ways we can draw a certain marble.
This exercise demonstrates a simple application of combinatorics, where we analyze possible results (i.e., Blue-Red, Red-Blue, Red-Red) and their respective probabilities.
Understanding this ensures a comprehensive idea of how different outcomes multiply together, contributing to the overall probability calculation.
Non-Replacement Probability
The concept of non-replacement probability specifically considers how the total number of items to draw from changes after each draw. This is due to not replacing the drawn item back into the pool.
In our exercise, this means the marble composition in the bag alters with each draw, affecting subsequent probabilities.
  • Initially, there are three marbles: 1 blue and 2 red. Each first draw outcome reduces this count by one, inevitably altering the next event's probabilities.
  • If a blue marble is drawn first (probability \(\frac{1}{3}\)), only red marbles remain, resulting in a subsequent 100% chance of drawing a red marble.
  • Conversely, a red marble first draw reduces the count to one blue and one red marble left, affecting the second draw with each having a \(\frac{1}{2}\) probability.
These adjustments highlight how non-replacement changes the scenarios, contrasting with replacement scenarios where each event probability would remain constant due to unchanged total count.

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

Taya is a contestant on a television quiz show. If she answers the next question correctly, she will win \(\$ 16,000\). If she misses the question, she will receive only \(\$ 1,000\). The question is multiple choice, and Taya has no idea what the correct answer is, so she will randomly choose one of the four answers. a. What is the expected value of Taya's earnings for the next question? b. If Taya can eliminate one answer and her probability of answering correctly is now one-third, what is the expected value?

A chili recipe calls for seven ingredients: ground beef, onions, beans, tomatoes, peppers, chili powder, and salt. There are no directions about the order in which the ingredients should be combined. You decide to add the ingredients in a random order. a. How many different arrangements are there? b. What is the probability that onions are first? c. What is the probability that the order is exactly as listed above? d. What is the probability that the order isn't exactly as listed above? e. What is the probability that beans are third?

This tree diagram shows possible results for the first two games in a three- game series between the Detroit Tigers and Texas Rangers. a. Copy and extend the diagram on your paper to show all outcomes of a three- game series. b. Highlight the path indicating that Texas won the first two games and Detroit won the final game. c. Does your diagram model permutations, combinations, or neither? Explain. d. If each outcome is equally likely, what is the probability that Texas won the first two games and Detroit won the third? (a) e. If you know Texas wins more than one game, what is the probability that the sequence is TTD?

The star hitter on the baseball team at City Community College had a batting average of \(.375\) before the start of a three-game series. (Note: Batting average is calculated by dividing hits by times at bat; sacrifice bunts and walks do not count as times at bat.) During the three games, he came to the plate to bat eleven times. In these eleven plate appearances, he walked twice and had one sacrifice bunt. He either got a hit or struck out in his other plate appearances. If his batting average was the same at the end of the three-game series as at the beginning, how many hits did he get?

Tom Fool wants to make his fortune playing the state lottery. To win the big prize, he must select the six winning numbers, which are drawn from the numbers 1 to 50 . He can select exactly six numbers for each ticket. a. How many sets of six different numbers from 1 to 50 are there? b. What is the probability that any one ticket will be a winner? c. If Tom buys 100 tickets each week, what is the probability that he wins in any one week? What is the probability that he loses in any one week? d. Draw a partial tree diagram of four weeks, and use the probability from \(11 \mathrm{c}\) to determine the probability that he will lose all four weeks. e. Determine the probability that he will lose every week for one year (52 weeks). f. At \(\$ 1\) a ticket, 100 tickets a week, 52 weeks a year, what is his cost?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.