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Conditional probability is a fundamental concept in probability theory that describes the likelihood of an event occurring, given that another event has already occurred. It is particularly useful in situations where we have some prior knowledge that influences the probability of an event.
To compute conditional probability, we use the formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Here, \( P(A|B) \) is the probability of event A occurring given event B has occurred, \( P(A \cap B) \) is the probability that both events A and B occur, and \( P(B) \) is the probability of event B.
For example, in the given exercise, to find the probability that a missing aircraft with an emergency locator is not discovered, we condition our calculation on additional information, such as how many aircraft with locators are usually discovered.
This helps us adjust our expectations more accurately by incorporating what we already know about related events.

The Law of Total Probability is a vital theorem in probability theory that relates marginal probabilities to conditional probabilities.
It allows us to compute the probability of a particular event based on several mutually exclusive cases that cover every possible scenario.
The law can be expressed as: \[ P(A) = \sum_{i} P(A|B_i) \cdot P(B_i) \] Here, the event \( A \) is expressed as the sum of its conditional probabilities over a set of disjoint events \( B_i \).
In our aircraft exercise, the probability of an aircraft having an emergency locator, \( P(E) \), is determined by considering both discovered and non-discovered scenarios.
By applying the law, we can determine every possible case where the aircraft might have an emergency locator, making our calculations more comprehensive and accurate.

The complement rule is a simple yet powerful principle in probability that helps us find the probability of an event not occurring.
It is based on the idea that every possible outcome is either the event itself or its complement. Hence, the probabilities of these two events add up to one.
The complement rule formula is: \[ P(A^c) = 1 - P(A) \] Where \( P(A^c) \) is the probability that event \( A \) does not occur.
In the context of the exercise, when we calculate the probability of discovering an aircraft without an emergency locator, we use the complement rule to find \( P(D^c) \), the probability of an aircraft not being discovered.
Thus, the complement rule provides a straightforward approach to finding probabilities of opposite or mutually exclusive events.