91Ó°ÊÓ

In probability, events are termed **independent** when the occurrence of one event does not affect the occurrence of another. For instance, when a construction firm bids on two separate contracts, the outcome of the bid on the first contract does not influence the outcome of the bid on the second. They operate independently.

When events are independent, the probability of both events occurring simultaneously can be found by multiplying the probabilities of each event. Mathematically, for two events, \(E_{1}\) and \(E_{2}\), we have:

• \(P(E_{1} \text{ and } E_{2}) = P(E_{1}) \times P(E_{2})\)

This concept simplifies computation because there is no need to consider additional factors.

In the given exercise, the probability of both bids being successful is determined by multiplying the probabilities of individual successes.

The **complement** of an event is essentially the probability of the event not occurring. This concept is crucial because it allows us to calculate probabilities indirectly.

If an event \(E\) has a probability \(P(E)\), then the probability of the complement of \(E\), denoted \(\text{not } E\), is given by:

• \(P(\text{not } E) = 1 - P(E)\)

This subtraction from one accounts for the fact that the total probability of all possible outcomes in any scenario is always equal to one.

In our exercise, this concept is used to find the probability of not securing either contract bid. By finding the complement probabilities of winning each bid, we then calculate the probability of not winning any bids at all.

**Joint probability** refers to the likelihood of two events occurring at the same time. When the events are independent, this probability can be computed by multiplying their individual probabilities. Hence, the joint probability of events \(E_{1}\) and \(E_{2}\) is depicted as \(P(E_{1} \cap E_{2})\).

However, if events are dependent, the joint probability would require considering the conditional probability affected by one event over the other, a scenario not applicable in this exercise.

In the context of our problem, where the construction firm's success on both bids is concerned, the joint probability \(P(E_{1} \cap E_{2})\) illustrates the likelihood of both outcomes occurring together. This is computed straightforwardly due to the independence of the events involved.

**Event probability** is the fundamental measure indicating the likelihood of a particular outcome occurring. It's expressed as a value between 0 and 1, where 0 implies impossibility, and 1 indicates certainty.

For any event \(E\), \(P(E)\) quantifies its probability, serving as a building block for more complex probability calculations, such as joint or complementary probabilities.

In this exercise, the probabilities of each bidding success \(P(E_{1})\) and \(P(E_{2})\) are given directly. These provide essential starting points for computing the joint probability of success on both bids, as well as the result of complementary probabilities to determine the chance of at least one bid failing.