Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

4. Probability

Addition Rule

Statistics

4. Probability

Addition Rule

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2:15 minutes

Problem 3.3.16

Textbook Question

16. Can Defects Of the cans produced by a company, 96% do not have a puncture, 93% do not have a smashed edge, and 89.3% have neither a puncture nor a smashed edge. Find
the probability that a randomly selected can does not have a puncture or a smashed edge.

Verified step by step guidance

Step 1: Define the events. Let A represent the event that a can does not have a puncture, and B represent the event that a can does not have a smashed edge. The probabilities given are P(A) = 0.96, P(B) = 0.93, and P(A ∩ B) = 0.893.

Step 2: Recall the formula for the union of two events. The probability that a can does not have a puncture or a smashed edge is given by P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Step 3: Substitute the given probabilities into the formula. Replace P(A) with 0.96, P(B) with 0.93, and P(A ∩ B) with 0.893 in the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Step 4: Simplify the expression. Add P(A) and P(B), then subtract P(A ∩ B) to calculate P(A ∪ B).

Step 5: Interpret the result. The value of P(A ∪ B) represents the probability that a randomly selected can does not have a puncture or a smashed edge.

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Key Concepts

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

Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this context, it helps quantify the chances of a can being free from defects, such as punctures or smashed edges. Understanding how to calculate probabilities is essential for solving the question.

Complement Rule

The Complement Rule states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring. This concept is crucial for determining the probability of a can not having a puncture or a smashed edge by using the given probabilities of the defects.

Union of Events

The Union of Events refers to the probability that at least one of multiple events occurs. In this case, it involves calculating the probability that a can has either a puncture or a smashed edge. Understanding how to combine probabilities of different events is key to finding the solution.

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Textbook Question

True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false,

explain why.

4. When two events are independent, they are also mutually exclusive.

Textbook Question

Graphical Analysis In Exercises 7 and 8, determine whether the events shown in the Venn diagram are mutually exclusive. Explain your reasoning.

Textbook Question

Recognizing Mutually Exclusive Events In Exercises 9–12, determine whether the events are mutually exclusive. Explain your reasoning.

10. Event A: Randomly select a student with a birthday in April.

Event B: Randomly select a student with a birthday in May.

Textbook Question

13. Students A physics class has 40 students. Of these, 12 students are physics majors and 16 students are minoring in math. Of the physics majors, three are minoring in math. Find the probability that a randomly selected student is minoring in math or a physics major.

Textbook Question

17. Selecting a Card A card is selected at random from a standard deck of 52 playing cards. Find the probability of each event.

c. Randomly selecting a 9 or a face card

Textbook Question

18. Rolling a Die You roll a die. Find the probability of each event.

b. Rolling a 2 or an odd number

Textbook Question

Mental Health A survey asks 4805 parents the severity of the mental issues they experienced from the coronavirus pandemic. The results are shown in the table. A parent is randomly selected from the sample. Find the probability of each event. (Adapted from Kaiser Family Foundation)

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c. The parent did not have major mental health issues or is a mother.

Textbook Question

25. Working from Home The table shows the results of a survey that asked 1811 people how often they work from home. A person is selected at random from the sample. Find the probability of each event.

b. The person is female or does not work from home.

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16. Can Defects Of the cans produced by a company, 96% do not have a puncture, 93% do not have a smashed edge, and 89.3% have neither a puncture nor a smashed edge. Findthe probability that a randomly selected can does not have a puncture or a smashed edge.

Key Concepts

Probability

Complement Rule

Union of Events

Watch next

16. Can Defects Of the cans produced by a company, 96% do not have a puncture, 93% do not have a smashed edge, and 89.3% have neither a puncture nor a smashed edge. Find
the probability that a randomly selected can does not have a puncture or a smashed edge.