Problem 142 - Applied Statistics and Probability for Engineers

Question

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

Step-by-Step Solution

Answer

Events A and B are not independent.

Step 1: Identify the Given Information

We are given: - Conditional probability: \(P(A \mid B) = 0.4\).- Probability of event B: \(P(B) = 0.8\).- Probability of event A: \(P(A) = 0.5\).

Step 2: Use the Independence Condition Formula

Two events \(A\) and \(B\) are independent if \(P(A \mid B) = P(A)\). Substitute the given values into this condition to check if \(0.4 = 0.5\).

Step 3: Compare Probabilities

Since \(P(A \mid B) = 0.4\) and \(P(A) = 0.5\), these two probabilities are not equal. This implies that \(A\) and \(B\) are not independent events.

Key Concepts

Conditional Probability

To understand conditional probability, consider it as the likelihood of an event occurring given that another event has already transpired. The conditional probability of event \(A\) given event \(B\) is denoted as \(P(A \mid B)\). It answers the question: "If I know that \(B\) has happened, what is the probability that \(A\) will also occur?" Thus, we calculate conditional probability using the formula:

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

In our exercise, \(P(A \mid B) = 0.4\). This means when event \(B\) happens, the probability that \(A\) also occurs is 40%. Conditional probability is sensitive to the given context, making it different from the standalone probability of an event. It's crucial for situations where past events or evidence modifies our evaluation of further possibilities.

Independence of Events

Events are considered independent if the occurrence of one does not affect the probability of the occurrence of the other. In mathematical terms, events \(A\) and \(B\) are independent if \(P(A \mid B) = P(A)\) or equivalently, \(P(B \mid A) = P(B)\).

• If these conditions hold true, the occurrence of event \(B\) has no impact on the probability of event \(A\), and vice versa.
• Independence is a key concept because it simplifies the mathematical modeling of multiple events happening at the same time.

In our problem, we need to determine if \(A\) and \(B\) are independent. We found \(P(A \mid B) = 0.4\) and \(P(A) = 0.5\). They aren't equal, indicating that event \(B\) affects the probability of \(A\), thus they are not independent. This reinforces the understanding that independence of events must align with these clear mathematical conditions.

Probability of Events

The probability of an event is a fundamental concept of probability theory. It measures the likelihood or certainty of an event occurring. Probabilities are values between 0 and 1, where 0 means the event will not occur and 1 means it certainly will.

• For example, the probability \(P(A) = 0.5\) indicates a 50% chance that event \(A\) will occur.
• Similarly, \(P(B) = 0.8\) indicates an 80% chance that event \(B\) will occur.

These individual probabilities provide a sense of expectation or prediction about the events based on the given conditions or observations. Understanding the probability of events enables effective decision-making and helps assess risks or chances in various real-world scenarios, such as games, forecasts, and decisions under uncertainty. In our context, knowing \(P(A)\) and \(P(B)\) helps evaluate conditional probabilities and assess independence.