91Ó°ÊÓ

Independent events are an essential concept in probability theory. When two events, say event A and event B, are independent, it means that the occurrence of one event does not influence the occurrence of the other. This is a key condition that allows us to apply specific probability rules.

Mathematically, this relationship is expressed by the fact that the probability of both events happening together (their intersection) is simply the product of their individual probabilities. In other words, if A and B are independent, then the formula is:

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

In the original exercise, with \( P(A) = 0.5 \) and \( P(B) = 0.2 \), we have calculated that \( P(A \cap B) = 0.1 \). This calculation stems from the independence of A and B, ensuring they do not affect each other's outcomes.

Understanding the union of events is critical for calculating the probability of either of two events occurring. The union of two events, A and B, noted as \( A \cup B \), represents the probability that either A occurs, B occurs, or both occur.

For independent events, the probability of their union can be found using the following formula:

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

The subtraction of \( P(A \cap B) \) accounts for any overlap in the events, avoiding double-counting situations where both A and B occur.

In the given problem, after calculating \( P(A \cap B) = 0.1 \), we concluded that \( P(A \cup B) = 0.6 \). This outcome helps in understanding how often at least one of these events would happen when considering them together.

The concept of complementary probability is rooted in understanding what doesn't happen when a certain event does occur. If you have an event A, the complement \( \bar{A} \) represents all outcomes that are not part of A. The probability of this complement is given by:

• \( P(\bar{A}) = 1 - P(A) \)

This approach ensures that the total probability between an event and its complement sums to 1, as it covers all possible outcomes.

In the exercise, we used this principle to determine \( P(\bar{A} \cap \bar{B}) \) and \( P(\bar{A} \cup \bar{B}) \). First, we found the probability where neither A nor B happen, \( P(\bar{A} \cap \bar{B}) = 0.4 \), by calculating the intersection of their complements. Then, using De Morgan's law, we determined \( P(\bar{A} \cup \bar{B}) \) simply by subtracting the union of A and B, demonstrating how complements properly fill out the full spectrum of event probability.