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Understanding the concept of relative frequency is essential when interpreting probabilities. When we say the probability of an event, such as a flight arriving on time, is 0.86, we are using the relative frequency interpretation of probability.

The relative frequency interpretation relates to the number of times an event occurs compared to the total number of trials. If we consider a large number of flights, the probability of 0.86 suggests that for every 100 flights we observe, approximately 86 of them are expected to arrive on time. It is important to note that this is an 'average' or an 'expectation' based on the probability value provided, and actual outcomes might vary.

This concept helps in making informed predictions about future events based on past occurrences. Students often find it easier to grasp probabilities when they can visualize them as the count of successes in a series of trials.

When dealing with probabilities, expressing them as fractions can be incredibly insightful. For example, the probability of the flight arriving on time is initially given as 0.86, a decimal. However, when we express this as a fraction, it becomes clearer how many out of a set number of trials result in the desired outcome.

In this instance, we convert the decimal to a fraction by considering the decimal as a part of 100. Thus, 0.86 becomes \(\frac{86}{100}\). This fraction represents the same idea as the decimal: out of 100 flights, we expect 86 to be on time. Using fractions can make it more tangible, especially when simplifying to the lowest terms to show the smallest possible 'group' that this probability might apply to.

Simplifying fractions is a valuable skill in mathematics, particularly in probability. It involves reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. To simplify the fraction \(\frac{86}{100}\), we find the greatest common divisor of both numbers, which is 2, and divide the numerator and denominator by this number.

After simplification, we get \(\frac{43}{50}\). This tells us that for every 50 occurrences (flights, in this context), we expect 43 successful outcomes (on-time arrivals). Students should note that simplifying fractions does not change the value of the probability; it only makes it easier to understand and often easier to work with in various probability problems.