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Probability Theory is a branch of mathematics that deals with the likelihood of different outcomes in uncertain situations. At its core, it uses the concept of a sample space, which encompasses all possible outcomes of a given scenario. Each outcome in the sample space can be assigned a probability, which is a number between 0 and 1, indicating how likely it is to occur.
By using probability, we can gain insights into the likelihood of various events, allowing for better decision making under uncertainty. In the context of the exercise, the company predicts the success of oil exploration projects using probability calculations.
This helps them to understand the risk and potential return of their investments. Basic probability operations involve events like union, intersection, and complements, which help in making such forecasts.

Complementary events are pairs of events in which one event happens if and only if the other does not happen. These are mutually exclusive and exhaustive events. In probability equations, if you have the probability of an event, you can easily find the probability of its complement.

• The formula to calculate the complement of an event is: \(P(A') = 1 - P(A)\).
• Here, \(A'\) is the event that \(A\) does not happen.

In the exercise, for instance, if the Asian project has a 40% chance of success, it means there's a 60% chance it will not succeed. Understanding complementary events allows businesses to prepare for unfavorable outcomes effectively.

Conditional Probability concerns the probability of an event occurring, given that another event has already happened. It's crucial in scenarios where events might be dependent or when considering a sequence of events.
The formula that expresses conditional probability is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
This reads as the probability of event A occurring given that B has occurred. In our exercise, the company might be interested in the success of the Asian project given information about success in general.
Determining conditional probabilities helps refine predictions by factoring in new, relevant information.

The Probability of Union of Events deals with the chance of at least one of multiple events occurring. Symbolically, this is represented as \(P(A \cup B)\), which denotes the probability of event A or event B, or both happening.
The formula for calculating this probability is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
In scenarios where the events are independent, as in our exercise, the formula simplifies by using the complements: \(P(A \cup B) = 1 - P(A' \cap B')\). For the company, knowing this probability helps in understanding the likelihood of success with either project.
This insight can guide decisions on how resources should be allocated to maximize the chances of at least one project's success.