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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 3.3.16
Chapter 3, Problem 3.3.16

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.

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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.
Recommended video:
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Introduction to Probability

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.
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Complementary Events

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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Probability of Multiple Independent Events
Related Practice
Textbook Question

"According to Bayes’ Theorem, the probability of event A , given that event B has occurred, is

P(A|B) = P(A) * P(B|A)P(A) * P(B|A) + P(A') * P(B|A').

In Exercises 33–38, use Bayes’ Theorem to find P(A|B).

34. P(A) = 3/8, P(A') = 5/8, P(B|A) = 2/3 , and P(B|A') = 3/5 "

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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, rewrite it as a true statement.

6. 7C5=7C2

Textbook Question

True or False? In Exercises 5 and 6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

6. If events A and B are dependent, then P(A and B) = P(A) · P(B).

Textbook Question

1. When you calculate the number of permutations of n distinct objects taken r at a time, what are you counting? Give an example.

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Textbook Question

"Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.

12. Not putting money in a parking meter and getting a parking ticket"

Textbook Question

"2. Give an example of

a. two events that are independent.

b. two events that are dependent."