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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 3.3.29
Chapter 3, Problem 3.3.29

29. Explain, in your own words, why in the Addition Rule for P(A or B or C), P(A and B and C) is added at the end of the formula.

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The Addition Rule for P(A or B or C) is used to calculate the probability of at least one of the events A, B, or C occurring. The formula accounts for the overlap between events to avoid overcounting.
When adding P(A), P(B), and P(C), the overlaps between pairs of events (e.g., P(A and B), P(A and C), P(B and C)) are included multiple times, so we subtract these pairwise intersections to correct for overcounting.
However, when we subtract the pairwise intersections, we inadvertently remove the probability of the triple intersection, P(A and B and C), too many times (once for each pair).
To correct this, we add P(A and B and C) back to the formula. This ensures that the probability of all three events occurring simultaneously is properly accounted for.
Thus, the term P(A and B and C) is added at the end of the formula to ensure the final probability calculation is accurate and avoids both overcounting and undercounting.

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Key Concepts

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

Addition Rule of Probability

The Addition Rule of Probability is a fundamental principle that calculates the probability of the occurrence of at least one of several events. It states that to find the probability of A or B or C occurring, you must sum the individual probabilities of each event and subtract the probabilities of their intersections to avoid double counting. This rule is essential for understanding how probabilities combine in scenarios involving multiple events.
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Probability of Non-Mutually Exclusive Events

Intersection of Events

The intersection of events refers to the scenario where two or more events occur simultaneously. In the context of the Addition Rule, P(A and B and C) represents the probability that all three events occur at the same time. This term is added at the end of the formula to ensure that the probability of all events happening together is included, preventing the underestimation of the total probability when events overlap.
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Probability of Multiple Independent Events

Overlapping Events

Overlapping events occur when two or more events share common outcomes, leading to potential double counting in probability calculations. In the Addition Rule, it is crucial to account for these overlaps by adding the intersection probabilities back into the total. This ensures that the final probability accurately reflects the likelihood of at least one of the events occurring without redundancy.
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Related Practice
Textbook Question

Warehouse In Exercises 51-54, a warehouse employs 24 workers on first shift, 17 workers on second shift, and 13 workers on third shift. Eight workers are chosen at random to be

interviewed about the work environment.

Find the probability of choosing four third-shift workers.

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Textbook Question

80. Unusual Events Can any of the events in Exercises 75-78 be considered unusual? Explain.

Textbook Question

In Exercises 7-14, perform the indicated calculation.

12. (10C7)/(14C7)

Textbook Question

23. Footrace There are 72 runners in a 10-kilometer race. How many ways can the runners finish first, second, and third?

Textbook Question

"Identifying Simple Events In Exercises 33-36, determine the number of outcomes in the event. Then decide whether the event is a simple event or not. Explain your reasoning.

36. You randomly select one card from a standard deck of 52 playing cards. Event B is selecting the ace of spades."

Textbook Question

3. What does the notation P(B|A) mean?