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Chapter 6: Problem 28

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 \(P(E)=.7, P(F)=.6\), and \(P(E \cap F)=.54\). 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 \(P(F \mid E)\).

Short Answer

Expert verified
The probability that the first airline's flight is fully booked given that the second airline's flight is fully booked is 0.9. The probability that the second airline's flight is fully booked given that the first airline's flight is fully booked is 0.7714.

Step by step solution

01

Define Notation

Start by defining the probability of each event and the intersection of these events. Here, \(P(E)=0.7\) is the probability of the first airline's flight being fully booked, \(P(F)=0.6\) is the probability of the second airline's flight being fully booked and \(P(E 鈭� F)=0.54\) is the probability of both flights being fully booked.
02

Calculate \(P(E \mid F)\)

To calculate the conditional probability, we need to apply the formula \(P(E \mid F) = P(E 鈭� F) / P(F)\). Inserting given probabilities, we get \(P(E \mid F) = 0.54 / 0.6 = 0.9\). Thus, the probability that the first airline's flight is fully booked given that the second airline's flight is fully booked is 0.9.
03

Calculate \(P(F \mid E)\)

Now, we calculate \(P(F \mid E)\) the probability that the second airline's flight is fully booked given that the first airline's flight is fully booked, following a similar method. We use the formula, \(P(F \mid E) = P(E 鈭� F) / P(E)\), yielding: \(P(F \mid E) = 0.54 / 0.7 = 0.7714\). Thus, the probability that the second airline's flight is fully booked given that the first airline's flight is fully booked is 0.7714.

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Key Concepts

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

Intersection of Events
In probability, the intersection of events describes the scenario where two or more events happen simultaneously. Imagine you're analyzing two flights from different airlines, like in our exercise. One is fully booked (event E), and the other is also fully booked (event F). The probability of both these events occurring together at the same time on a particular day is expressed as \(P(E \cap F)\).
To better understand this concept:
  • The intersection focuses on joint outcomes 鈥� here, both flights being fully booked.
  • If events E and F are independent, the occurrence of one does not influence the other. However, we often deal with dependent events in practice, where knowing one event happened changes the probability of the other.
  • In our problem, \(P(E \cap F) = 0.54\). That's the chance both airlines鈥� flights are fully booked on the same day.
Understanding intersections helps us when we delve into more complex probability calculations, like conditional probability.
Probability of Events
Probability helps us measure the chance of a specific outcome occurring. In our exercise, we're dealing with simple events and their probabilities. For instance, \(P(E) = 0.7\) reflects the likelihood of the first airline's flight being fully booked. Similarly, \(P(F) = 0.6\) is the likelihood of the second airline鈥檚 flight meeting the same fate.
Some key points to consider about probabilities of events:
  • Each event's probability ranges from 0 to 1, where 0 indicates no chance and 1 means certainty.
  • For mutually exclusive events, however, their intersection (or joint probability) is zero.
  • But in our case, the events are not mutually exclusive because there exists a non-zero probability (\(P(E \cap F)\)) where both flights are fully booked.
Using these basic probabilities sets the stage for understanding how events interact, especially when conditioned on each other.
Conditional Probability Formula
The Conditional Probability Formula allows us to compute the probability of one event given the occurrence of another. This is especially useful when we know two events may interact or be dependent on each other.The formula is expressed as:\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]This formula calculates the probability of event A happening, given event B has happened. In the exercise, we calculated:
  • \(P(E \mid F)\), where E is the first airline鈥檚 flight being fully booked, given the second flight (event F) is fully booked:
  • Inserting values gives \(P(E \mid F) = 0.54 / 0.6 = 0.9\). This means there's a 90% chance that the first flight is fully booked if the second is as well.
  • Similarly, for \(P(F \mid E)\), we substitute the respective values: \(P(F \mid E) = 0.54 / 0.7 = 0.7714\). Meaning, if the first flight is fully booked, the second has a 77.14% chance of being fully booked too.
Understanding the conditional probability formula allows for complex predictive insights in everyday situations, such as forecast planning based on known occurrences.

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Most popular questions from this chapter

The Associated Press (San Luis Obispo TelegramTribune, August 23,1995 ) reported on the results of mass screening of schoolchildren for tuberculosis (TB). For Santa Clara County, California, the proportion of all tested kindergartners who were found to have TB was .0006. The corresponding proportion for recent immigrants (thought to be a high-risk group) was .0075. Suppose that a Santa Clara County kindergartner is selected at random. Are the events selected student is a recent immigrant and selected student has \(T B\) independent or dependent events? Justify your answer using the given information.

A theater complex is currently showing four R-rated movies, three \(\mathrm{PG}-13\) movies, two \(\mathrm{PG}\) movies, and one \(\mathrm{G}\) movie. The following table gives the number of people at the first showing of each movie on a certain Saturday: $$ \begin{array}{rlc} \text { Theater } & \text { Rating } & \begin{array}{l} \text { Number of } \\ \text { Viewers } \end{array} \\ \hline 1 & \mathrm{R} & 600 \\ 2 & \mathrm{PG}-13 & 420 \\ 3 & \mathrm{PG}-13 & 323 \\ 4 & \mathrm{R} & 196 \\ 5 & \mathrm{G} & 254 \\ 6 & \mathrm{PG} & 179 \\ 7 & \mathrm{PG}-13 & 114 \\ 8 & \mathrm{R} & 205 \\ 9 & \mathrm{R} & 139 \\ 10 & \mathrm{PG} & 87 \\ \hline \end{array} $$Suppose that a single one of these viewers is randomly selected. a. What is the probability that the selected individual saw a PG movie? b. What is the probability that the selected individual saw a PG or a PG-13 movie? c. What is the probability that the selected individual did not see an R movie?

Suppose that a box contains 25 bulbs, of which 20 are good and the other 5 are defective. Consider randoml selecting three bulbs without replacement. Let \(E\) denote the event that the first bulb selected is good, \(F\) be the event that the second bulb is good, and \(G\) represent the event that the third bulb selected is good. a. What is \(P(E)\) ? b. What is \(P(F \mid E)\) ? c. What is \(P(G \mid E \cap F)\) ? d. What is the probability that all three selected bulbs are good?

An individual is presented with three different glasses of cola, labeled C, D, and P. He is asked to taste all three and then list them in order of preference. Suppose that the same cola has actually been put into all three glasses. a. What are the simple events in this chance experiment, and what probability would you assign to each one? b. What is the probability that \(\mathrm{C}\) is ranked first? c. What is the probability that \(\mathrm{C}\) is ranked first \(a n d \mathrm{D}\) is ranked last?

Of the 60 movies reviewed last year by two critics on their joint television show, Critic 1 gave a "thumbs-up" rating to 15 , Critic 2 gave this rating to 20 , and 10 of the movies were rated thumbs-up by both critics. Suppose that 1 of these 60 movies is randomly selected. a. Given that the movie was rated thumbs-up by Critic 1 , what is the probability that it also received this rating from Critic \(2 ?\) b. If the movie did not receive a thumbs-up rating from Critic 2, what is the probability that it also did not receive a thumbs up rating from Critic \(1 ?\) (Hint: Construct a table with two rows for the first critic [for "up" and "not up"] and two columns for the second critic: then enter the relevant probabilities.)
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