Question

If \(C_{1}, \ldots, C_{k}\) are \(k\) events in the sample space \(\mathcal{C}\), show that the probability that at least one of the events occurs is one minus the probability that none of them occur; i.e., $$P\left(C_{1} \cup \cdots \cup C_{k}\right)=1-P\left(C_{1}^{c} \cap \cdots \cap C_{k}^{c}\right)$$

Short answer

The stated equation \(P(C_{1} \cup \cdots \cup C_{k}) = 1 - P(C_{1}^{c} \cap \cdots \cap C_{k}^{c})\) is proven by using the complement rule, concepts of union and intersection of events, and the definition of not occurrence of all events equating to at least an occurrence in the sets.

Step-by-step solution

Understanding concepts

Understand the notation used - \(C_i\) represents any event from the sample space and \(C_i^c\) represents the complement of the event \(C_i\) which means that \(C_i\) does not occur. The union \(\cup\) of events implies that at least one of the events is occurring, while the intersection \(\cap\) implies that all the events are occurring simultaneously.

Complement Rule

Use the complement rule - the probability of an event not happening is 1 minus the probability of that event happening. In this context, \(1 - P(C_1^c \cap \cdots \cap C_k^c)\) represents the probability of the event that not all sets from \(C_1\) to \(C_k\) do not occur. If this event does not happen, it means that at least one of the events \(C_1\) to \(C_k\) is occurring, which is \(C_1 \cup \cdots \cup C_k\).

Final step

Finally, since the not occurrence of all events \(C_1\) to \(C_k\) equates to at least one occurrence in the sets from \(C_1\) to \(C_k\), it infers that \(P(C_1 \cup \cdots \cup C_k) = 1 - P(C_1^c \cap \cdots \cap C_k^c)\), hence proving the stated equation.

Key concepts

Complement Rule

In probability theory, the complement rule is a fundamental concept used to find the probability of an event not happening. Understanding this rule is essential, especially when dealing with complex events or multiple events. Simply put, if the probability of an event \(A\) happening is denoted as \(P(A)\), then the probability of \(A\) not happening, also known as its complement, is \(1 - P(A)\). This rule utilizes the fact that the total probability of all possible outcomes in a sample space is always equal to 1.

For example, consider a single event \(C_i\) in our sample space. The complement of \(C_i\), represented as \(C_i^c\), signifies the scenario where \(C_i\) does not occur. The probability of the complement, \(P(C_i^c)\), would therefore be \(1 - P(C_i)\). The complement rule is very useful because calculating the probability of an event not happening is often simpler than calculating the probability directly for some complex structures in events.

In the context of combining multiple events, such as \(C_1, C_2, \ldots, C_k\), applying the complement rule allows you to efficiently determine the probability of none of these events occurring by observing:\[1 - P(C_1^c \cap C_2^c \cap \cdots \cap C_k^c)\] This statement reflects the complementary nature of the probability that all events \(C_1, C_2, \ldots, C_k\) happen.

Union of Events

The union of events is a critical idea in probability, represented by the symbol \(\cup\). It embodies the concept of combining multiple events such that at least one of the events occurs. Consider two events \(A\) and \(B\). The union of \(A\) and \(B\), denoted as \(A \cup B\), refers to the scenario where either event \(A\), event \(B\), or both occur. It's an inclusive event that accounts for all possibilities where any number of specified events take place.

In cases where more than two events are considered, such as with groups \(C_1, C_2, \ldots, C_k\), the union \(C_1 \cup C_2 \cup \cdots \cup C_k\) encompasses occurrences where at least one of these events takes place. Thus, the union of events is a way to broaden the scope of probability calculations, allowing researchers and students to assess the likelihood of numerous possible outcomes across various scenarios.

Understanding unions becomes particularly powerful when paired with the complement rule, as in our original exercise. Here, by finding the complement of all events happening, it becomes easier to find the exact probability that at least one event from \(C_1, C_2, \ldots, C_k\) will actually occur.

Probability of Events

In essence, the probability of an event represents the likelihood that the event will happen. It is quantified as a value between 0 and 1 inclusive, where 0 means the event will never occur, and 1 denotes certainty of occurrence. Calculating the probability of events lays the foundation for making predictions and informed decisions based on data.

To evaluate the probability of any event, say \(E\), within a given sample space, we utilize the fundamental probability formula: \[P(E) = \frac{\text{Number of favorable outcomes for E}}{\text{Total number of possible outcomes in the sample space}}\] This formula aids in straightforward calculations, especially for single events with finite outcomes.

When evaluating multiple events or more sophisticated occurrences, we often utilize rules like the complement rule and the union principle. These enable the calculation of probabilities of collective events happening, such as in the original exercise where the probability of a union of several events was determined using the complement of their intersection's probability. By understanding and applying these probability principles, dealing with complex events becomes manageable, whether in academic exercises or real-world applications.