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Chapter 2: Problem 122

If \(P(A \mid B)=0.4, P(B)=0.8,\) and \(P(A)=0.5,\) are the events \(A\) and \(B\) independent?

Short Answer

Expert verified
The events are not independent.

Step by step solution

01

Understanding the Problem

We are given \(P(A \mid B) = 0.4\), \(P(B) = 0.8\), and \(P(A) = 0.5\). We need to determine if events \(A\) and \(B\) are independent. Events \(A\) and \(B\) are independent if \(P(A \mid B) = P(A)\).
02

Compute and Compare Probabilities

The problem states that \(P(A \mid B) = 0.4\) and \(P(A) = 0.5\). For \(A\) and \(B\) to be independent, these should be equal. \(P(A \mid B)\) is not equal to \(P(A)\).
03

Conclude About Independence

Since \(P(A \mid B) = 0.4\) is not equal to \(P(A) = 0.5\), we conclude that events \(A\) and \(B\) are not independent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Conditional probability is all about the likelihood of an event occurring given that another event has already occurred. This concept is usually referred to as "the probability of A given B" and is denoted as \(P(A \mid B)\). It's like asking, "What are the chances it's raining (Event A), given that we see people holding umbrellas (Event B)?"
  • The formula for calculating conditional probability is \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \), assuming \(P(B) > 0\).
  • It shows the relationship between two events and how the probability of one influences the other.
In the context of the given exercise, the value \(P(A \mid B) = 0.4\) states the probability of event A happening if B has occurred. It's crucial as it is used to check whether two events, A and B, are independent.
Event Independence
Event independence can be a tricky idea, but it's very significant in probability. Two events, A and B, are said to be independent if the occurrence of one does not affect the probability of the other. In simpler terms, knowing that one event has happened doesn鈥檛 give any information about the likelihood of the other event happening.
  • The mathematical definition of independence is \( P(A \mid B) = P(A) \).
  • If they are independent, the joint probability can also be expressed as \(P(A \cap B) = P(A) \cdot P(B)\).
In our problem statement, we compare \(P(A \mid B) = 0.4\) with \(P(A) = 0.5\). Since they aren't equal, A and B aren't independent. Independence implies no influence between events. Here, knowing B occurred clearly alters the probability of A.
Probability Comparison
When comparing probabilities, it's all about seeing how one probability stands in relation to another. This exercise tasked us with checking if \(P(A \mid B)\) is equal to \(P(A)\).
  • If they match, it indicates independence, meaning the occurrence of B doesn't affect A's likelihood.
  • Different values imply some level of dependency. Here, \(P(A \mid B) = 0.4\) and \(P(A) = 0.5\)
The discrepancy indicates that observing B changes our prediction about A happening, disproving independence. Comparing these provides insight into how events relate in probability terms. Observing these differences is key to understanding the relationship between two events.

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Most popular questions from this chapter

An inspector working for a manufacturing company has a \(99 \%\) chance of correctly identifying defective items and a \(0.5 \%\) chance of incorrectly classifying a good item as defective. The company has evidence that its line produces \(0.9 \%\) of nonconforming items. (a) What is the probability that an item selected for inspection is classified as defective? (b) If an item selected at random is classified as nondefective, what is the probability that it is indeed good?

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