91Ó°ÊÓ

In probability theory, the complement rule is a fundamental concept that helps us determine the probability of an event not happening. It is a simple yet powerful tool when we know the probability of an event occurring. When dealing with probabilities, remember that the total probability for all possible outcomes must add up to 1.
This means if we have an event, let's call it "Event A," the probability that Event A does not occur is the complement of the probability that it does occur. Mathematically, we can express this using the formula:

• The probability of an event not occurring: \( P(A^c) = 1 - P(A) \)

In the problem with the trucks, we used the complement rule to find the probability that neither truck was available by calculating the complement of at least one truck being available. Specifically, we found that the probability neither truck was available is given by the expression:

• \( 1 - P(A_1 \cup A_2) \)

This approach uses the logic that either the event of at least one truck being available happens or it doesn’t, and these are the only two possible outcomes.

The Addition Rule for Probabilities is used when we wish to find the probability of either one of several events occurring. It combines probabilities of multiple events, providing the probability that at least one of the events will occur.
When events overlap or have elements in common, this rule helps account for that overlap. The formula for the addition rule is:

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

Here, \( P(A \cup B) \) refers to the probability of either event A or event B (or both) occurring. \( P(A) \) and \( P(B) \) are the probabilities of each event occurring individually, and \( P(A \cap B) \) is the probability that both events occur together.
In the example of the trucks, to determine the probability that at least one truck was available, we used the values provided to calculate \( P(A_1 \cup A_2) \):

• \( P(A_1 \cup A_2) = 0.75 + 0.50 - 0.30 = 0.95 \)

This calculation helped us identify the possibility that either or both trucks were in service.

Understanding events and probabilities is crucial in solving real-world problems through probability theory. An "event" is an outcome or a set of outcomes from an experiment or situation. When we measure the likelihood of events, we are often tasked with determining how probable it is for different scenarios to occur.
For example, suppose we have two service trucks, and we're interested in their availability. Here we identify two events:

• Event \( A_1 \): The first truck is available.
• Event \( A_2 \): The second truck is available.

Each event is associated with a probability that quantifies how likely it is to happen. To solve the problem with the trucks, we were given:

• \( P(A_1) = 0.75 \)
• \( P(A_2) = 0.50 \)
• The combined event \( P(A_1 \cap A_2) = 0.30 \)

The task was then to manipulate these probabilities to uncover the likelihood of another configuration of events: the scenario where neither truck was available. By applying known probability rules, like the complement rule and addition rule, we can derive probabilities for these complex scenarios efficiently.