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Chapter 5: Problem 2

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

Expert verified
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

01

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.
02

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 鈥� let's say 100 flights 鈥� 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 鈥� in this case, an airline flight arriving on time 鈥� 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.

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Most popular questions from this chapter

Suppose events \(E\) and \(F\) are mutually exclusive with \(P(E)=0.14\) and \(P(F)=0.76\) i. What is the value of \(P(E \cap F) ?\) ii. What is the value of \(P(E \cup F)\) ? b. Suppose that for events \(A\) and \(B, P(A)=0.24, P(B)=0.24\), and \(P(A \cup B)=0.48 .\) Are \(A\) and \(B\) mutually exclusive? How can you tell?

A Nielsen survey of teens between the ages of 13 and 17 found that \(83 \%\) use text messaging and \(56 \%\) use picture messaging (鈥淗ow Teens Use Media," Nielsen, June 2009). Use these percentages to explain why the two events \(T=\) event that a randomly selected teen uses text messaging and \(P=\) event that a randomly selected teen uses picture messaging cannot be mutually exclusive.

There are two traffic lights on Shelly's route from home to work. Let \(E\) denote the event that Shelly must stop at the first light, and define the event \(F\) in a similar manner for the second light. Suppose that \(P(E)=0.4, P(F)=0.3\), and \(P(E \cap F)=0.15 .\) (Hint: See Example 5.5) a. Use the given probability information to set up a "hypothetical \(1000 "\) table with columns corresponding to \(E\) and not \(E\) and rows corresponding to \(F\) and not \(F\). b. Use the table from Part (a) to find the following probabilities: i. the probability that Shelly must stop for at least one light (the probability of \(E \cup F)\). ii. the probability that Shelly does not have to stop at either light. iii. the probability that Shelly must stop at exactly one of the two lights. iv. the probability that Shelly must stop only at the first light.

The following statement is from a letter to the editor that appeared in USA Today (September 3,2008 ): "Among Notre Dame's current undergraduates, our ethnic minority students \((21 \%)\) and international students \((3 \%)\) alone equal the percentage of students who are children of alumni (24\%). Add the \(43 \%\) of our students who receive need-based financial aid (one way to define working-class kids), and more than \(60 \%\) of our student body is composed of minorities and students from less affluent families." Do you think that the statement that more than \(60 \%\) of the student body is composed of minorities and students from less affluent families is likely to be correct? Explain.

An appliance manufacturer offers extended warranties on its washers and dryers. Based on past sales, the manufacturer reports that of customers buying both a washer and a dryer, \(52 \%\) purchase the extended warranty for the washer, \(47 \%\) purchase the extended warranty for the dryer, and \(59 \%\) purchase at least one of the two extended warranties. a. Use the given probability information to set up a "hypothetical 1000 " table. b. Use the table from Part (a) to find the following probabilities: i. the probability that a randomly selected customer who buys a washer and a dryer purchases an extended warranty for both the washer and the dryer. ii. the probability that a randomly selected customer purchases an extended warranty for neither the washer nor the dryer.
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