91Ó°ÊÓ

In probability, understanding independent events is crucial. Independent events are those where the occurrence of one event does not affect the probability of the other. For instance, consider two coin flips. Each flip is independent of the previous one, meaning the result of the first flip has no effect on the outcome of the second flip. The same rule applies when selecting people for jury duty across different years in the given exercise. If a person is selected one year, it does not influence whether they'll be selected in subsequent years.

Here's why: Candidates are chosen randomly, and previous selections do not alter their chances in future draws. This randomness and lack of influence from previous events confirm the independence of each year's selection. Thus, when calculating the probability of sequential selections, we assume independence.

The multiplication rule is a key principle in calculating probabilities for independent events. This rule states that if you want to find the probability of two (or more) independent events happening together, you multiply their individual probabilities.

Consider an example where we want to know the probability of rolling a 3 on two consecutive rolls with a fair six-sided die. Each independent roll has a probability of \( \frac{1}{6} \). By applying the multiplication rule, you multiply these probabilities: \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \).

In the exercise, the probability of a person being chosen for jury duty in any single year is 0.15. To find the probability of being selected for two consecutive years, you'd calculate: \( 0.15 \times 0.15 = 0.0225 \). If you extend this to three years, you continue multiplying 0.15 for each additional year: \( 0.15 \times 0.15 \times 0.15 = 0.003375 \).

This simplicity and clarity in the process make the multiplication rule highly useful for handling multiple independent events.

Probability calculation often begins with understanding the total number of outcomes and desired outcomes. In simple terms, probability is the chance or likelihood of an event happening among all possible outcomes.

The formula to calculate basic probability is: \[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

For example, if the probability of being selected for jury duty is given as 15%, it means there is a 0.15 probability each year. To determine more complex scenarios, such as being selected for consecutive years, you'd utilize both the concept of independent events and the multiplication rule to refine these probability calculations. In the given scenario, calculating the probability over multiple years involved understanding how individual yearly probabilities contribute to the overall likelihood of being chosen consecutively. By handling such calculations carefully, you not only arrive at the correct answer but also enhance your comprehension of probability theory as a whole.