91Ó°ÊÓ

In probability, complementary events are pairs of events that together encompass all possible outcomes within a sample space. They are quite fundamental in probability theory since the sum of the probabilities of an event and its complement always equals 1. This stems from the idea that one event either happens or it does not, covering all possible scenarios.

For example, if we have an event "A" with probability \( P(A) = 0.3 \), the complement, which is "not A", is represented as \( A^c \) and would have a probability \( P(A^c) = 1 - P(A) = 0.7 \).

In our exercise, we explored complementary events when we calculated \( P((H \text{ AND } G)^c) \) and \( P((H \text{ OR } G)^c) \). Here, we determined the probability of each event not happening. The formula used for complements, \( P(A^c) = 1 - P(A) \), is a straightforward way to view probabilities from an alternative perspective.

Joint probability refers to the probability of two or more events occurring simultaneously. It is denoted as \( P(A \text{ AND } B) \). This concept is essential when evaluating the likelihood of multiple events happening at the same time.

In the context of our problem, we calculated the joint probability \( P(H \text{ AND } G) \) as given, which is 0.14. This means there is a 14% chance that both events \( H \) and \( G \) will occur at the same time.

The joint probability is crucial for solving problems involving the overlap of events, as it helps differentiate between separate occurrences (just one event happening) and joint occurrences (both events happening together). It is often used in conjunction with other principles like the addition rule for probabilities, which helps calculate the probability of the union of events, as is done in our exercise.

The union of events in probability, denoted as \( A \text{ OR } B \), refers to the likelihood that at least one of several events occurs. This encompasses scenarios where one or both events happen, capturing the broader spectrum of possible outcomes.

The formula used to calculate the union of events \( P(A \text{ OR } B) \) is \[ P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B) \] to account for any overlap where both events might occur simultaneously.

In the given exercise, using the provided values, we calculated it as \( P(H \text{ OR } G) = 0.55 \). This shows a 55% probability that either event \( H \) or event \( G \) or both occur. Understanding the union of events is critical in scenarios where the focus is on possible points of success across multiple events, and is a common method for complex probability calculations across various fields.