Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 25m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

4. Probability

Basic Concepts of Probability

Statistics

4. Probability

Basic Concepts of Probability

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1:24 minutes

Problem 3.1.3

Textbook Question

3. Explain why the statement is incorrect: The probability of rain is 150%.

Verified step by step guidance

Step 1: Begin by explaining the concept of probability. Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1 (or equivalently, 0% to 100%). A probability of 0 means the event is impossible, while a probability of 1 (or 100%) means the event is certain.

Step 2: Clarify that probabilities cannot exceed 1 (or 100%). This is because probability represents a fraction of the total possible outcomes, and the total probability of all possible outcomes must sum to 1 (or 100%).

Step 3: Discuss why the statement 'The probability of rain is 150%' is incorrect. A probability of 150% exceeds the maximum possible value of 100%, which violates the fundamental definition of probability.

Step 4: Provide an example to reinforce the concept. For instance, if the probability of rain were 50%, it would mean there is a 50/100 chance of rain occurring. A probability of 150% would imply more than certainty, which is not logically or mathematically valid.

Step 5: Conclude by emphasizing that probabilities are always constrained within the range of 0 to 1 (or 0% to 100%), and any value outside this range is not meaningful in the context of probability theory.

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Key Concepts

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

Probability Range

Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1, or as a percentage from 0% to 100%. A probability of 0% means the event will not happen, while 100% means it is certain to occur. Therefore, stating a probability of 150% exceeds this range and is mathematically invalid.

Understanding Percentages

Percentages represent a fraction of 100. When discussing probabilities, they indicate how likely an event is to occur out of 100 possible outcomes. A probability of 150% suggests that the event is more than certain, which is impossible, as it cannot exceed the totality of possible outcomes.

Interpretation of Weather Forecasts

Weather forecasts often express the probability of rain as a percentage, indicating the chance of rain occurring in a specific area over a given time. A probability of rain is based on historical data and meteorological models, and it should always fall within the 0% to 100% range to be meaningful and accurate.

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Textbook Question

2. Determine whether each number could represent the probability of an event. Explain your reasoning. a. 25/25

Textbook Question

2. Determine whether each number could represent the probability of an event. Explain your reasoning. b. 333.3%

Textbook Question

2. Determine whether each number could represent the probability of an event. Explain your reasoning. c. 2.3

Textbook Question

2. Determine whether each number could represent the probability of an event. Explain your reasoning. d. -0.0004

Textbook Question

True or False? In Exercises 7-10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

9. A probability of 1/10 indicates an unusual event.

Textbook Question

Matching Probabilities In Exercises 11-16, match the event with its probability.

a. 0.95

b. 0.005

c. 0.25

d. 0

e. 0.375

f. 0.5

11. A random number generator is used to select a number from 1 to 100. What is the probability of selecting the number 153?

Textbook Question

Matching Probabilities In Exercises 11-16, match the event with its probability.

a. 0.95

b. 0.005

c. 0.25

d. 0

e. 0.375

f. 0.5

14. A game show contestant must randomly select a door. One door doubles her money while the other three doors leave her with no winnings. What is the probability she selects the

door that doubles her money?

Textbook Question

Identifying the Sample Space of a Probability Experiment In Exercises 25-32, identify the sample space of the probability experiment and determine the number of outcomes in the sample space. Draw a tree diagram when appropriate.

25. Guessing the initial of a student's middle name

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