91Ó°ÊÓ

Probability is a way to measure how likely an event is to occur. In our problem, we have two events: event \( A \) (where the first component functions properly) and event \( B \) (where the second component functions properly). Each event has its own probability. For instance, the probability that the second component functions throughout its design life is denoted as \( P(B) = 0.9 \). This means there is a 90% chance that the second component will work satisfactorily.

• The probability of an event is a value between 0 and 1.
• A probability of 0 means the event will not occur.
• A probability of 1 indicates the event will definitely occur.

In this exercise, we are also analyzing the combined probabilities of events. Specifically, the problem gives us \( P(A \cap B) = 0.75 \), the probability that both components work, and \( P(A \cup B) = 0.96 \), meaning either or both components work. Understanding these values helps us find the conditional probability, which is the chance of one event occurring given another event has already occurred.

In probability, understanding how events relate is critical. "Union" and "intersection" are two key concepts that describe these relationships.
The "union" \((A \cup B)\) of two events refers to the probability that at least one of the events occurs. In mathematical terms, it's calculated using the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] This formula accounts for any overlap between events, preventing us from counting occurrences twice. In our exercise, the probability of the union, \( P(A \cup B) \), is given as 0.96, or 96%. This means one or both components function satisfactorily with this probability.

The "intersection" \((A \cap B)\) of two events is the probability of both events occurring simultaneously. As given, \( P(A \cap B) = 0.75 \). This shows a 75% chance that both components are functioning well.

• The intersection informs us about simultaneous occurrences.
• The union gives a broader view of any occurrence among events.

By using these concepts, we unravel complex relationships between variables.

To solve the given problem of conditional probability steps are followed:
1. **Identifying Given Probabilities** Start by noting all the probabilities provided in the problem: \( P(B) = 0.9 \), \( P(A \cup B) = 0.96 \), and \( P(A \cap B) = 0.75 \). This sets the groundwork for solving the problem.

2. **Understanding and Setting Up** The goal is to find \( P(B|A) \), the probability of the second component working given the first is working.

3. **Using Conditional Probability Formula** The formula for conditional probability is: \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] This ratio provides a way to adjust the likelihood of \( B \) happening, considering \( A \) has already occurred.

4. **Finding \( P(A) \) with Union Rule** Determine \( P(A) \) using the union formula: \[ P(A) = P(A \cup B) + P(A \cap B) - P(B) \] Substitute to find \( P(A) = 0.81 \).

5. **Calculating and Deriving Final Answer** Substitute \( P(A \cap B) = 0.75 \) and \( P(A) = 0.81 \) into the conditional probability formula: \[ P(B|A) = \frac{0.75}{0.81} \] This results in approximately \( P(B|A) \approx 0.9259 \).
6. **Conclusion** The result shows there's about a 92.59% probability that the second component functions well, knowing the first component does.