Question

An airline reports that for a particular flight operating daily between Phoenix and Atlanta, the probability of an on-time arrival is \(0.86 .\) Give a relative frequency interpretation of this probability.

Short answer

A relative frequency interpretation of the probability \(0.86\) for an on-time arrival is that if the airline operates this particular flight 100 times, then we would expect about 86 of those flights to arrive on time.

Step-by-step solution

Translate Probability to Relative Frequency

The given probability of an on-time arrival is \(0.86 .\) This means that for every single flight, there's an \(86\%\) chance of arriving on time. To give a relative frequency interpretation, we can think about what this would look like over a large number of flights. Relative frequency is essentially the proportion of times an event occurs in a large number of trials or occurrences.

Calculate Frequency

To translate this into a relative frequency representation, we'll assume that the event of on-time arrival occurs \(86\%\) of the time over a large number of flights. Let's use 100 flights as an example for our large number of trials. Since the probability of an on-time arrival is \(0.86 .\), over these 100 flights, we would expect about 86 of them to arrive on time.

Key concepts

Relative Frequency

When we talk about the relative frequency of an event, we are referring to the number of times that event occurs compared to the total number of trials. The concept is often used in statistics to provide an empirical probability that complements theoretical calculations.

In the context of our airline example, if the probability of an on-time arrival is given as 0.86, this suggests that if we observe the flight over a large number of separate occasions — let's say 100 flights — we would see that approximately 86 of these flights arrive on time. This assumes that the conditions remain consistent across each flight. The relative frequency is therefore the observed frequency (86 on-time arrivals) divided by the total number of observed instances (100 flights). As such, the relative frequency provides a tangible way to understand probability through observed data.

Probability Calculation

Probability calculation involves determining the likelihood of an event occurring. In a mathematical sense, it is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 signifies certainty. Probability can be derived from relative frequencies or through theoretical models.

For example, the probability for on-time arrival being 0.86 can be viewed as a conversion of the relative frequency into a more abstract, numerical representation. Probabilities can be multiplied, added, or combined with other probabilities to calculate the likelihood of complex series of events, taking care to use the appropriate rules for independent or dependent events. The clarity of probability lies in its ability to provide a standardized metric for comparing the likelihoods of various outcomes.

On-Time Arrival Probability

The on-time arrival probability specifically refers to the chance that a particular event — in this case, an airline flight arriving on time — will occur. This is of great interest not only to passengers but also to airline companies, as it reflects operational efficiency.

Given a probability of 0.86 for on-time arrival, it implies that the airline is confident that out of a sequence of flights, a high percentage will arrive within the scheduled time. To put it plainly, if you were to frequently travel on this specific flight, about 8-9 times out of 10, you can expect to reach your destination on time. These probabilities are typically based on historical data and are crucial for logistics, scheduling, and managing customer expectations.