91Ó°ÊÓ

When dealing with independent events, understanding the intersection of these events is crucial. The intersection, denoted by \(A \cap B\), represents the probability of both events \(A\) and \(B\) occurring simultaneously. For independent events, the key is that one event does not affect the probability of the other occurring. This is why for independent events, the probability of the intersection can be simply calculated by multiplying their individual probabilities. This is expressed in the formula:

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

In our specific case, we have \(P(A) = 0.4\) and \(P(B) = 0.2\). By multiplying these values, we get:

• \( P(A \cap B) = 0.4 \times 0.2 = 0.08 \)

This result signifies that there is an 8% chance of both, events \(A\) and \(B\), happening together.

Conditional probability refers to the probability of an event occurring, given that another event has already occurred. It is denoted as \( P(A \mid B) \) which reads as the probability of \(A\) given \(B\). Even though \(A\) and \(B\) are indeed independent, we can still calculate this conditional probability. The formula used for conditional probability is:

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

In our exercise, we already know that the probability of \(A \cap B\) is 0.08 and \(P(B) = 0.2\). By substituting these values into our formula, we find:

• \( P(A \mid B) = \frac{0.08}{0.2} = 0.4 \)

This result demonstrates that even after event \(B\) has occurred, the probability of \(A\) occurring remains the same at 40%, aligning with the evidence that events \(A\) and \(B\) are indeed independent.

The addition rule for probability is used to find the probability of the union of two events, represented as \( P(A \cup B) \). The union denotes either event \(A\) or event \(B\) or both occurring. For independent events, the addition rule states:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

This formula accounts for the overlap (intersection) of probabilities, ensuring we don't double-count the occurrence when both events happen together. In the context of our problem, we simply input the values we have:

• \( P(A) = 0.4 \)
• \( P(B) = 0.2 \)
• \( P(A \cap B) = 0.08 \)
• Substituting these, we get: \( P(A \cup B) = 0.4 + 0.2 - 0.08 = 0.52 \)

Thus, the probability of either event \(A\), event \(B\), or both occurring is 52%. This helps us understand how these probabilities interrelate when combined.