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Chapter 5: Problem 41

Airline tickets can be purchased online, by telephone, or by using a travel agent. Passengers who have a ticket sometimes don't show up for their flights. Suppose a person who purchased a ticket is selected at random. Consider the following events: \(O=\) event selected person purchased ticket online \(N=\) event selected person did not show up for flight Suppose \(P(O)=0.70, P(N)=0.07,\) and \(P(O \cap N)=0.04\) a. Are the events \(N\) and \(O\) independent? How can you tell? b. Construct a hypothetical 1000 table with columns corresponding to \(N\) and \(\operatorname{not} N\) and rows corresponding to \(O\) and \(\operatorname{not} O\). c. Use the table to find \(P(O \cup N)\). Give a relative frequency interpretation of this probability.

Short Answer

Expert verified
a. The events N and O are not independent, as \(P(N \cap O) \ne P(N) * P(O)\). b. The hypothetical 1000 table: ``` | N | not N | Total --------------------------- O | 40 | 660 | 700 --------------------------- not O| 30 | 270 | 300 --------------------------- Total| 70 | 930 | 1000 ``` c. The probability of passengers who either purchased a ticket online or did not show up for their flight, \(P(O \cup N)= 0.73\). Out of every 1000 passengers, approximately 730 of them will either purchase a ticket online or not show up for their flight.

Step by step solution

01

Determine if events N and O are independent

To find out if events N and O are independent, we will use the given probabilities and the formula mentioned above: \(P(N \cap O) = P(N) * P(O)\) We have the values for: - \(P(O) = 0.70\) - \(P(N) = 0.07\) - \(P(O \cap N) = 0.04\) Now, let's apply these values to the formula: \(0.04 = 0.07 * 0.70\) \(0.04 = 0.049\) Since the \(P(N \cap O) \ne P(N) * P(O)\), events N and O are not independent.
02

Construct a hypothetical 1000 table

To create the table, we will use the percentage probabilities given and multiply them by 1000: - Passengers who purchased a ticket online and did not show up (O 鈭� N) = 40 passengers - Passengers who purchased a ticket online (O) = 700 passengers - Passengers who did not show up for their flight (N) = 70 passengers Now we can find the remaining passengers who did not purchase a ticket online (not O) and showed up for their flight (not N). Total passengers - 1000. - Passengers who did not purchase a ticket online and showed up (not O 鈭� not N) = 960 - 700 = 260 passengers - Passengers who did not purchase a ticket online (not O) = 300 passengers Our table will look like this: ``` | N | not N | Total --------------------------- O | 40 | 660 | 700 --------------------------- not O| 30 | 270 | 300 --------------------------- Total| 70 | 930 | 1000 ```
03

Find the probability of \(P(O \cup N)\) and provide a relative frequency interpretation

To find the probability of passengers who either purchased a ticket online or did not show up for their flight (O 鈭� N), we will use the following formula: \(P(O \cup N) = P(O) + P(N) - P(O \cap N)\) From the given probabilities: - \(P(O) = 0.70\) - \(P(N) = 0.07\) - \(P(O \cap N) = 0.04\) Applying these values to the formula: \(P(O \cup N) = 0.70 + 0.07 - 0.04\) \(P(O \cup N) = 0.73\) Hence, the probability of passengers who either purchased a ticket online or did not show up for their flight is 0.73 or 73%. In terms of a relative frequency interpretation, out of every 1000 passengers, approximately 730 of them will either purchase a ticket online or not show up for their flight.

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Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics that deals with studying random events and measuring the likelihood that these events will occur. It provides a systematic framework for quantifying and reasoning about uncertainty. In probability theory, the probability of an event is a number between 0 and 1. A probability of 0 indicates that an event will not happen, while a probability of 1 indicates that an event will certainly happen.
\[P(A)\] is the standard notation used to represent the probability of event A occurring. The layout of certain probabilities can be simple, but when combined with multiple events, it can become more complex. Events can be independent, dependent, exclusive, or non-exclusive, which affects how probabilities are calculated.
Probability theory is frequently applied in various fields such as finance, insurance, science, and engineering, helping professionals to make informed decisions under uncertainty. A fundamental aspect of probability theory is understanding the concepts of joint, marginal, and conditional probabilities.
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred. This concept is essential for understanding dependencies between events in probability theory. Mathematically, the conditional probability of event A given event B is represented as \(P(A|B)\) and is calculated as follows:
\[P(A|B) = \frac{P(A \cap B)}{P(B)},\] provided \(P(B) eq 0\).
Conditional probability allows us to update the probability of an event based on new information. Understanding conditional probability is crucial in many real-world situations, such as assessing risks in healthcare, making decisions under uncertainty, and managing stocks and portfolios.
  • **Independent Events**: If two events, A and B, are independent, the probability of A given B is just the probability of A. In other words, \(P(A|B) = P(A)\), which means the occurrence of event B does not influence the probability of event A.
  • **Dependent Events**: For dependent events, the occurrence of one affects the probability of the other.
Hypothetical Table
A hypothetical table is a simple yet effective tool used to organize and display information in probability problems. It helps in visualizing the relationships between different events and simplifies the process of calculating probabilities. In our example, a hypothetical table with 1000 total entries was constructed based on given probabilities.
The table provided separate categories, such as those who purchased a ticket online (O), did not show up (N), and those who did each of these or neither. This organization makes it easier to understand the distribution of events:
  • **Cell Values**: For example, 40 people purchased a ticket online and did not show up.
  • **Total Values**: In the problem, 1000 represents the total sample set.
The hypothetical table thus serves as a helpful visualization aid, allowing us to quickly sum relevant sections to find probabilities of combined events, such as \(P(O \cup N)\). Using such tables, we can provide interpretations based on the calculated values and give insights into the likelihood of various combinations of events.

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Most popular questions from this chapter

The article "Scrambled Statistics: What Are the Chances of Finding Multi-Yolk Eggs?" (Significance [August 2016]: 11) gives the probability of a double-yolk egg as 0.001 . a. Give a relative frequency interpretation of this probability. b. If 5000 eggs were randomly selected, about how many double-yolk eggs would you expect to find?

In a January 2016 Harris Poll, each of 2252 American adults was asked the following question: "If you had to choose, which ONE of the following sports would you say is your favorite?" ("Pro Football is Still America's Favorite Sport," www.theharrispoll.com/sports/Americas_Fav_Sport_2016. html, retrieved April 25,2017 ). Of the survey participants, \(33 \%\) chose pro football as their favorite sport. The report also included the following statement, "Adults with household incomes of \(\$ 75,000-<\$ 100,000(48 \%)\) are especially likely to name pro football as their favorite sport, while love of this particular game is especially low among those in \(\$ 100,000+$$ households \)(21 \%)\( Suppose that the percentages from this poll are representative of American adults in general. Consider the following events: \)F=\( event that a randomly selected American adult names pro football as his or her favorite sport \)L=\( event that a randomly selected American has a household income of \)\$ 75,000-<\$ 100,000\( \)H=\( event that a randomly selected American has a household income of \)\$ 100,000+$$ b. Are the events \(F\) and \(L\) mutually exclusive? Justify your answer. c. Are the events \(H\) and \(L\) mutually exclusive? Justify your answer. d. Are the events \(F\) and \(H\) independent? Justify your answer.

A construction firm bids on two different contracts. Let \(E_{1}\) be the event that the bid on the first contract is successful, and define \(E_{2}\) analogously for the second contract. Suppose that \(P\left(E_{1}\right)=0.4\) and \(P\left(E_{2}\right)=0.3\) and that \(E_{1}\) and \(E_{2}\) are independent events. a. Calculate the probability that both bids are successful (the probability of the event \(E_{1}\) and \(E_{2}\) ). b. Calculate the probability that neither bid is successful (the probability of the event \(\operatorname{not} E_{1}\) and not \(E_{2}\) ). c. What is the probability that the firm is successful in at least one of the two bids?

A rental car company offers two options when a car is rented. A renter can choose to pre-purchase gas or not and can also choose to rent a GPS device or not. Suppose that the events \(A=\) event that gas is pre-purchased \(B=\) event that a GPS is rented are independent with \(P(A)=0.20\) and \(P(B)=0.15\). a. Construct a hypothetical 1000 table with columns corresponding to whether or not gas is pre-purchased and rows corresponding to whether or not a GPS is rented. b. Use the table to find \(P(A \cup B)\). Give a long-run relative frequency interpretation of this probability.

5.59 Two different airlines have a flight from Los Angeles to New York that departs each weekday morning at a certain time. Let \(E\) denote the event that the first airline's flight is fully booked on a particular day, and let \(F\) denote the event that the second airline's flight is fully booked on that same day. Suppose that \(P(E)=0.7, P(F)=0.6\) and \(P(E \cap F)=0.54\). a. Calculate \(P(E \mid F),\) the probability that the first airline's flight is fully booked given that the second airline's flight is fully booked. b. Calculate \(P(F \mid E)\).
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