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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.R.4
Chapter 3, Problem 3.R.4

"In Exercises 1-4, identify the sample space of the probability experiment and determine the number of outcomes in the event. Draw a tree diagram when appropriate.
4. Experiment: Guessing the gender(s) of the three children in a family
Event: Guessing that the family has two boys"

Verified step by step guidance
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Step 1: Understand the problem. The experiment involves guessing the genders of three children in a family. Each child can either be a boy (B) or a girl (G), and we are tasked with identifying the sample space and determining the number of outcomes where the family has exactly two boys.
Step 2: Define the sample space. The sample space represents all possible combinations of genders for the three children. Since each child can independently be either a boy or a girl, the sample space can be represented as: {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}.
Step 3: Count the total number of outcomes in the sample space. Since there are 2 possible outcomes (B or G) for each of the 3 children, the total number of outcomes is calculated as 2^3 = 8.
Step 4: Identify the outcomes in the event. The event is that the family has exactly two boys. To find these outcomes, look for combinations in the sample space where exactly two of the three children are boys. These outcomes are: {BBG, BGB, GBB}.
Step 5: Count the number of outcomes in the event. From the identified outcomes in Step 4, there are 3 outcomes where the family has exactly two boys. Optionally, you can draw a tree diagram to visualize the branching possibilities for each child (B or G) and highlight the relevant outcomes.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Space

The sample space is the set of all possible outcomes of a probability experiment. In the context of guessing the genders of three children, the sample space includes all combinations of boys (B) and girls (G), such as BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. Understanding the sample space is crucial for calculating probabilities and analyzing events.
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Sampling Distribution of Sample Proportion

Event

An event is a specific outcome or a set of outcomes from the sample space that we are interested in. In this case, the event is guessing that the family has two boys. This event can be represented by the outcomes BBG, BGB, and GBB from the sample space, which helps in determining the probability of this event occurring.
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Tree Diagram

A tree diagram is a visual representation used to illustrate all possible outcomes of a probability experiment. Each branch of the tree represents a possible choice or outcome, leading to further branches for subsequent choices. For the experiment of guessing the genders of three children, a tree diagram can help visualize the combinations of boys and girls, making it easier to count outcomes and understand the structure of the sample space.
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Related Practice
Textbook Question

In Exercises 49-53, use counting principles to find the probability.

51. A shipment of 200 calculators contains 3 defective units. What is the probability that a sample of three calculators will have

c. at least one defective calculator?

Textbook Question

In Exercises 45-48, use combinations and permutations.

46. Five players on a basketball team must each choose one of the five players on the opposing team to defend. In how many ways can the players choose their defensive assignments?

Textbook Question

"In Exercises 5 and 6, use the Fundamental Counting Principle.

5. A student must choose from seven classes to take at 8:00 A.M., four classes to take at 9:00 A.M., and three classes to take at 10:00 A.M. How many ways can the student arrange the schedule?"

Textbook Question

"In Exercises 1-4, identify the sample space of the probability experiment and determine the number of outcomes in the event. Draw a tree diagram when appropriate.

1. Experiment: Tossing four coins

Event: Getting three heads"

Textbook Question

In Exercises 49-53, use counting principles to find the probability.

52. A class of 40 students takes a statistics exam. The results are shown in the table at the left. Three students are selected at random. What is the probability that

b. all three students received a C or better?

Textbook Question

In Exercises 49-53, use counting principles to find the probability.

53. A corporation has six male senior executives and four female senior executives. Four senior executives are chosen at random to attend a technology seminar. What is the

probability of choosing

c. two men and two women?