Problem 143 - Applied Statistics and Probability for Engineers

Question

If \(P(A \mid B)=0.3, P(B)=0.8,\) and \(P(A)=0.3,\) are the events \(B\) and the complement of \(A\) independent?

Step-by-Step Solution

Answer

Yes, the events \(B\) and \(A^c\) are independent.

Step 1: Understand the Complement Rule

The complement of event \(A\), denoted as \(A^c\), represents the event that \(A\) does not occur. Given that \(P(A) = 0.3\), the probability of \(A^c\) is \(P(A^c) = 1 - P(A) = 1 - 0.3 = 0.7\). We will use this in conjunction with \(P(B)\) to check for independence.

Step 2: Recall Definition of Independence

Two events \(B\) and \(A^c\) are considered independent if \(P(B) \times P(A^c) = P(B \cap A^c)\). We need to verify this equality to determine if \(B\) and \(A^c\) are independent.

Step 3: Use Bayes' Theorem for Probability of Intersection

We know \(P(A \,|\, B) = 0.3\). Using the formula for conditional probability, \[P(A \,|\, B) = \frac{P(A \cap B)}{P(B)}\]\, we find the probability of intersection: \[P(A \cap B) = P(A \,|\, B) \times P(B) = 0.3 \times 0.8 = 0.24.\] Hence, \[P(A^c \cap B) = P(B) - P(A \cap B) = 0.8 - 0.24 = 0.56.\]

Step 4: Check for Independence of B and A^c

Using the values calculated in previous steps, check if \(P(B) \times P(A^c) = P(B \cap A^c)\).Substitute the values: \[0.8 \times 0.7 = 0.56.\]Since \(0.56 = 0.56\), the equality holds, indicating that the events \(B\) and \(A^c\) are independent.

Key Concepts

Complement Rule

Understanding the concept of the complement is crucial in probability theory. The complement of an event, labeled as \(A^c\), denotes all outcomes that are not part of event \(A\). For example, if event \(A\) represents rolling a die and getting a 4, the complement \(A^c\) would be rolling anything other than a 4. This means that the probabilities of these two events add up to 1. Mathematically, it is expressed as:

• \(P(A^c) = 1 - P(A)\)

If \(P(A) = 0.3\), then \(P(A^c) = 0.7\). Knowing the complement helps in simplifying many problems, especially when checking for independence, as seen in the original exercise.

Independent Events

Two events, say \(A\) and \(B\), are independent if the occurrence of one does not affect the probability of the other occurring. In simpler terms, the probability of \(A\) happening is the same whether \(B\) happens or not. The formal definition states:

• \(P(A \cap B) = P(A) \times P(B)\)

This means the joint probability of \(A\) and \(B\) occurring is simply the product of their individual probabilities. Hence, if this condition holds true, \(A\) and \(B\) are independent. Conversely, if it does not, the events are dependent. In the step-by-step solution, this concept is applied to check if the events \(B\) and \(A^c\) are independent by verifying if \(P(B \cap A^c)\) equals \(P(B) \times P(A^c)\).

Conditional Probability

Conditional probability refers to the probability of event \(A\) occurring given that \(B\) has already occurred. It is represented mathematically as \(P(A \mid B)\).The formula for conditional probability is:

• \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\)

This formula helps us calculate the likelihood of \(A\) happening when \(B\) is known to occur. It's particularly useful when considering sequences of events or understanding dependencies.Using the given problem, the probability \(P(A \mid B) = 0.3\) is utilized to determine \(P(A \cap B)\) by rearranging the formula:

• \(P(A \cap B) = P(A \mid B) \times P(B)\)

This calculation provides the joint probability that both \(A\) and \(B\) occur, which is an essential step to check for the independence of events.

Bayes' Theorem

Bayes' Theorem is a fundamental result in probability theory. It describes the probability of an event based on prior knowledge of related events. Specifically, it allows us to update our beliefs in light of new evidence. The theorem is expressed as:

• \(P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)}\)

Although Bayes' Theorem is not directly used in the exercise, it provides the foundation for conditional probability calculations and the overall framework for understanding probability relationships between events.In practical applications, Bayes' Theorem is widely used in statistical inference and decision-making processes, allowing for precise probability updates as new information becomes available. It's a powerful tool for transforming abstract mathematical concepts into real-world probabilities.