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In probability theory, understanding the concept of independent events is fundamental. Independent events are two or more events where the occurrence of one event does not affect the occurrence of another. In simpler terms, the outcome of one event has no impact on the outcome of another. For instance, flipping a coin and rolling a die are independent events because the result of the coin flip does not influence the result of the die roll.

Applying this to the context of our exercise where individuals are called for jury duty, we can say that being selected for jury duty in one year is an independent event from being selected in another year. This is because the selection in one year does not affect the selection in the following year, given that the same individual cannot be selected more than once in a calendar year.

It's worth noting that identifying events as independent is critical before we can use certain rules to calculate the combined probability of these events. If events were not independent, we would need to approach the problem differently, perhaps with rules concerning conditional probabilities or dependent events.

The probability calculation involves finding the likelihood of an event occurring. This is often expressed as a fraction or decimal between 0 and 1, where 0 indicates that the event will never occur, and 1 indicates certainty that the event will occur. For example, if the probability of an event is 0.5, this means there is a 'fifty-fifty' chance of the event occurring.

In our exercise, the probability that an eligible person is selected for jury duty in one year is given as 15%, which translates to 0.15 when expressed as a decimal. To comprehend probability calculations, one must convert percentages to decimals as such and understand that a 15% chance of occurring is the same as saying the event will occur 15 times out of 100 trials on average.

When we speak about calculating the probability of multiple independent events occurring sequentially, such as being selected for jury duty multiple years in a row, we move beyond simple probability calculation and into the realm of combined probabilities, which is governed by the multiplication rule.

The multiplication rule for independent events is a vital tool in probability. It allows us to calculate the probability of two or more independent events occurring together by multiplying their separate probabilities. The formula for this rule is:\[ P(A \cap B) = P(A) \cdot P(B) \]where \( P(A \cap B) \) is the probability of both events \( A \) and \( B \) occurring, and \( P(A) \) and \( P(B) \) are the probabilities of each event occurring separately. This multiplication rule only applies when the events are independent, as in our jury duty scenario.

When we apply this rule to our example, we discover the combined probability of being selected for jury duty for consecutive years. For two years, the calculation would be \( 0.15 \cdot 0.15 \), and for three years, it would be \( 0.15 \cdot 0.15 \cdot 0.15 \). Thus, the multiplication rule simplifies the process of finding probabilities for a series of independent events, making it an indispensable component in a wide array of probability problems.

Consequently, grasping the multiplication rule is essential for anyone dealing with probability, because it frequently serves as the foundation for more complex probability theories and applications.