Language: Complement of “At Least One” Let A=the event of getting...

4. Probability

Complements

Statistics

4. Probability

Complements

1:57 minutes

Problem 4.3.1

Textbook Question

Language: Complement of “At Least One” Let A=the event of getting at least one defective calculator when four are randomly selected with replacement from a batch. Write a statement describing event A

Verified step by step guidance

Step 1: Understand the problem. The event A is defined as 'getting at least one defective calculator' when four calculators are randomly selected with replacement. This means we are interested in the complement of the event where all four calculators are non-defective.

Step 2: Recall the complement rule in probability. The complement of an event A is the event that A does not occur. Mathematically, P(A) = 1 - P(A').

Step 3: Define the complement of event A. The complement of event A (denoted as A') is the event where none of the four calculators selected are defective. This means all four calculators are non-defective.

Step 4: Write a statement describing event A. Event A can be described as 'at least one of the four calculators selected is defective.' This is equivalent to saying 'not all four calculators are non-defective.'

Step 5: Use the complement rule to calculate probabilities if needed. To find the probability of event A, you would calculate P(A) = 1 - P(A'), where P(A') is the probability that all four calculators are non-defective.

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Key Concepts

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

Complementary Events

In probability, the complement of an event A is the event that A does not occur. For example, if A is the event of getting at least one defective calculator, the complement would be the event of getting no defective calculators at all. Understanding complementary events is crucial for calculating probabilities, as the sum of the probabilities of an event and its complement equals one.

Probability with Replacement

When selecting items from a batch with replacement, each selection is independent of the others. This means that the probability of selecting a defective calculator remains constant for each of the four selections. This concept is essential for accurately calculating the likelihood of events when the same item can be chosen multiple times.

At Least One Probability

The phrase 'at least one' refers to the probability of an event occurring one or more times in a series of trials. To find this probability, it is often easier to calculate the complement (the probability of the event not occurring at all) and subtract it from one. This approach simplifies the calculation, especially in scenarios involving multiple trials.

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