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In probability theory, independent events are those that do not influence each other. This means that the occurrence of one event has no impact on the likelihood of the other event occurring. Using this principle, we can consider events like tossing a coin or rolling a die, where the outcome of one does not affect the outcome of another. The independence concept allows us to make separate calculations for each event without worrying about their mutual impact.
This is crucial in probability calculations because it simplifies the computation of complex problems. If two events are independent, calculating the combined probability is straightforward and is governed by a specific rule called the multiplication rule. By understanding that events are independent, we can use this special property to ease our calculations and make accurate probability predictions.

The multiplication rule is a fundamental principle used to determine the probability of multiple independent events occurring together. To use this rule, you simply multiply the probabilities of each event involved. The formula for this is straightforward:

• If Event A has a probability of P(A) and Event B has a probability of P(B),
• Then the probability of both events happening, denoted as P(A and B), is given by the formula:

\[P(A \text{ and } B) = P(A) \times P(B)\]This approach makes use of the independence between events, ensuring that one event's outcome does not alter the probability of the other. Consider the example of our generator and power failure scenario. Each event has its own probability, and since they are independent, applying the multiplication rule becomes possible. This turns into an efficient way to see how likely it is for two separate events to occur simultaneously.

Calculating the probability of multiple events, especially those that are independent, requires careful application of the multiplication rule. When we have two or more independent events, like in the power and generator scenario, we want to know how likely it is for them all to occur together.
For instance, in a situation where there is a 30% chance of a power failure and a 9% chance that the backup generator will fail during a snowstorm, the probability of both failing is the product of their individual probabilities. By following this approach, we can determine how multiple scenarios play out in tandem. This strategy not only helps in academic scenarios but is also highly valuable in real-life decision-making and risk assessment. Understanding the probability of multiple events is fundamental in fields ranging from engineering to finance.

To calculate the probability of two independent events both occurring, we need to understand and apply the steps involved thoroughly. Knowing that the individual probabilities are distinct and do not impact each other is key.
Step 1: Identify the probability of each individual event. Here, it means noting down the probabilities given: the power failure has a probability of 0.30, and the generator failure has a probability of 0.09.

• Power failure P(A) = 0.30
• Generator failure P(B) = 0.09

Step 2: Multiply these probabilities to find the probability of both events occurring. This step involves applying the multiplication rule for independent events as discussed earlier:\[P(A \text{ and } B) = P(A) \times P(B) = 0.30 \times 0.09 = 0.027\]This calculated probability, 0.027, represents the likelihood that both the primary and backup power systems will fail during a snowstorm. Mastering event probability calculation assists in evaluating various outcomes and making informed choices when dealing with uncertainties.