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When dealing with probabilities, complementary probability is an essential concept to understand. It refers to the probability of the opposite event occurring. Simply put, if you know the probability of an event, the complementary probability tells you how likely it is that the event won’t happen. The sum of the probability of an event and its complement is always 1. This is because something will either happen or it won't—there's no middle ground.
For example, if the probability of a person deciphering a code is \( \frac{1}{5} \), the probability of them not deciphering it is \( 1 - \frac{1}{5} = \frac{4}{5} \). Complementary probabilities are particularly useful when calculating complex events, as they allow you to focus on the scenarios where the event does not occur.
Understanding and using complementary probabilities can simplify many probability questions. Instead of figuring out every detail, you can determine the likelihood of the opposite scenario and subtract it from 1.

In probability theory, independent events are those events whose occurrence or outcome does not affect the probability of the other events. This means, if Event A occurs, it has no bearing on whether Event B occurs, and vice versa.
Independence is a common assumption that simplifies the calculation of probabilities. In the exercise, the probability of each person deciphering the message is independent of the others. Each person has their own separate probability of success \( \frac{1}{5}, \frac{1}{4}, \) and \( \frac{1}{3} \), respectively.
Understanding whether events are independent is crucial because it informs how probabilities are calculated. If events are dependent, then the occurrence of one affects the occurrence of the other, and different rules apply.
In cases where events are independent, calculating probabilities can often involve multiplication, as each event's probability remains constant regardless of the others.

The probability multiplication rule is used to find the joint probability of multiple independent events occurring. For independent events, the rule states that the probability of all events occurring is the product of their individual probabilities. This is because each event occurs without influencing the others.
For example, in the exercise, the goal is to find the probability that all individuals fail to decipher the message. We calculate this by multiplying the probabilities of each person failing: \( \frac{4}{5} \times \frac{3}{4} \times \frac{2}{3} \). Once calculated, this gives the overall probability of all three failing together.
This rule is particularly helpful for complex problems with multiple events, as it simplifies the process of finding the probability of concurrent independent events.

• Always ensure the events are truly independent before applying the rule.
• The rule can dramatically streamline solving probability problems by breaking them into manageable parts.