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Chapter 6: Problem 99

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

Short Answer

Expert verified
The probability that the viewer saw a PG movie, a PG or a PG-13 movie, and did not see an R movie are calculated as per the steps detailed above. Mathematical calculations need to be carried out to arrive at the final probabilities.

Step by step solution

01

Calculate Total Viewers

It's important to first calculate the total number of viewers across all theatres. Add up all the numbers from the 'Number of Viewers' column to get this total.
02

Calculate Probabilities for Part A

For Part A, total up the number of viewers for PG rated movies. Divide this total by the total number of viewers calculated in Step 1. This will be the probability that a viewer selected at random saw a PG movie.
03

Calculate Probabilities for Part B

For Part B, sum up the number of viewers for both PG and PG-13 rated movies. Divide this total by the total number of viewers obtained in Step 1. This will be the probability that a viewer selected at random saw either a PG or a PG-13 movie.
04

Calculate Probabilities for Part C

For Part C, add up the number of viewers for R rated movies. Divide this number by the total number of viewers obtained in Step 1 to get the probability of a viewer watching an R rated movie. Subtract this result from 1 to find the probability that a viewer did not watch an R rated movie.

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Key Concepts

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

Random Selection Probability
Understanding random selection probability is crucial for making predictions in various scenarios, such as drawing the name of a winner from a hat or determining the likelihood of a certain outcome in a population. In the context of our exercise, random selection probability involves the chance of a single individual being selected from a group based on certain characteristics鈥攊n this case, the movie rating they watched.

For instance, to find the probability that the chosen individual watched a PG movie, we sum the viewership of all the PG movies and divide it by the total number of viewers. This calculation assumes that each viewer has an equal chance of being selected. Applying this to larger populations, random selection probability allows researchers, businesses, and policymakers to make informed decisions by understanding the likelihood of events among diverse groups of individuals.
Movie Rating Viewership
Movie rating viewership can provide insight into audience preferences and behaviors. By reviewing the number of people attending movies of different ratings, businesses can tailor their offerings to meet demand, and movie creators can understand which content reaches larger audiences. Our textbook problem involves a breakdown of viewers by movie rating, which is a typical data set in entertainment market research.

Viewership data helps in creating a profile of what types of movies are popular among certain demographics. For example, understanding that a substantial proportion of theatergoers are watching PG-13 movies could prompt more productions within this rating. The exercise teaches how to calculate viewership proportions, which is a fundamental concept in both market research and probability.
Statistical Analysis
Statistical analysis encompasses a variety of methods used to interpret, infer, and predict information from data. The exercise requires us to perform basic statistical analysis by first determining the total data set size and then calculating the proportion of cases that meet specific criteria.

In our example, calculating the total viewers across all movie categories provides a foundational figure for further analysis. When we calculate the probabilities for the various movie ratings, we are, in essence, conducting a form of descriptive analysis. This form of analysis is commonly used to summarize and describe the features of a data set in a manageable form. Knowing the technique for calculating the probability of non-R rated movie viewership, for instance, could translate to understanding market segments in a business context or evaluating the prevalence of a health outcome in public health.

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Most popular questions from this chapter

In an article that appears on the web site of the American Statistical Association (www.amstat.org), Carlton Gunn, a public defender in Seattle, Washington, wrote about how he uses statistics in his work as an attorney. He states: I personally have used statistics in trying to challenge the reliability of drug testing results. Suppose the chance of a mistake in the taking and processing of a urine sample for a drug test is just 1 in 100 . And your client has a "dirty" (i.e., positive) test result. Only a 1 in 100 chance that it could be wrong? Not necessarily. If the vast majority of all tests given- say 99 in \(100-\) are truly clean, then you get one false dirty and one true dirty in every 100 tests, so that half of the dirty tests are false. Define the following events as \(T D=\) event that the test result is dirty, \(T C=\) event that the test result is clean, \(D=\) event that the person tested is actually dirty, and \(C=\) event that the person tested is actually clean. a. Using the information in the quote, what are the values of \mathbf{i} . ~ \(P(T D \mid D)\) iii. \(P(C)\) ii. \(P(T D \mid C)\) iv. \(P(D)\) b. Use the law of total probability to find \(P(T D)\). c. Use Bayes' rule to evaluate \(P(C \mid T D)\). Is this value consistent with the argument given in the quote? Explain.

Refer to the following information on births in the United States over a given period of time: $$ \begin{array}{lr} \text { Type of Birth } & \text { Number of Births } \\ \hline \text { Single birth } & 41,500,000 \\ \text { Twins } & 500,000 \\ \text { Triplets } & 5,000 \\ \text { Quadruplets } & 100 \\ \hline \end{array} $$ Use this information to approximate the probability that a randomly selected pregnant woman who reaches full term a. Delivers twins b. Delivers quadruplets c. Gives birth to more than a single child

The events \(E\) and \(T_{j}\) are defined as \(E=\) the event that someone who is out of work and actively looking for work will find a job within the next month and \(T_{i}=\) the event that someone who is currently out of work has been out of work for \(i\) months. For example, \(T_{2}\) is the event that someone who is out of work has been out of work for 2 months. The following conditional probabilities are approximate and were read from a graph in the paper "The Probability of Finding a Job" (American Economic Review: Papers \& Proceedings [2008]: \(268-273\) ) $$ \begin{array}{ll} P\left(E \mid T_{1}\right)=.30 & P\left(E \mid T_{2}\right)=.24 \\ P\left(E \mid T_{3}\right)=.22 & P\left(E \mid T_{4}\right)=.21 \\ P\left(E \mid T_{5}\right)=.20 & P\left(E \mid T_{6}\right)=.19 \\ P\left(E \mid T_{7}\right)=.19 & P\left(E \mid T_{8}\right)=.18 \\ P\left(E \mid T_{9}\right)=.18 & P\left(E \mid T_{10}\right)=.18 \\ P\left(E \mid T_{11}\right)=.18 & P\left(E \mid T_{12}\right)=.18 \end{array} $$ a. Interpret the following two probabilities: i. \(\quad P\left(E \mid T_{1}\right)=.30\) ii. \(\quad P\left(E \mid T_{6}\right)=.19\) b. Construct a graph of \(P\left(E \mid T_{i}\right)\) versus \(i\). That is, plot \(P\left(E \mid T_{i}\right)\) on the \(y\) -axis and \(i=1,2, \ldots, 12\) on the \(x\) -axis. c. Write a few sentences about how the probability of finding a job in the next month changes as a function of length of unemployment.

A friend who works in a big city owns two cars, one small and one large. Three-quarters of the time he drives the small car to work, and one-quarter of the time he takes the large car. If he takes the small car, he usually has little trouble parking and so is at work on time with probability .9. If he takes the large car, he is on time to work with probability .6. Given that he was at work on time on a particular morning, what is the probability that he drove the small car?

Medical insurance status-covered (C) or not covered (N) - is determined for each individual arriving for treatment at a hospital's emergency room. Consider the chance experiment in which this determination is made for two randomly selected patients. The simple events are \(O_{1}=(\mathrm{C}, \mathrm{C})\), meaning that the first patient selected was covered and the second patient selected was also covered, \(O_{2}=(\mathrm{C}, \mathrm{N}), O_{3}=(\mathrm{N}, \mathrm{C})\), and \(O_{4}=\) \((\mathrm{N}, \mathrm{N}) .\) Suppose that probabilities are \(P\left(O_{1}\right)=.81\), \(P\left(O_{2}\right)=.09, P\left(O_{3}\right)=.09\), and \(P\left(O_{4}\right)=.01\). a. What outcomes are contained in \(A\), the event that at most one patient is covered, and what is \(P(A)\) ? b. What outcomes are contained in \(B\), the event that the two patients have the same status with respect to coverage, and what is \(P(B)\) ?
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