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Chapter 6: Problem 31

The article "Chances Are You Know Someone with a Tattoo, and He's Not a Sailor" (Associated Press, June 11, 2006) included results from a survey of adults aged 18 to 50 . The accompanying data are consistent with summary values given in the article. $$ \begin{array}{l|cc} & \text { At Least One Tattoo } & \text { No Tattoo } \\ \hline \text { Age 18-29 } & 18 & 32 \\ \text { Age 30-50 } & 6 & 44 \\ \hline \end{array} $$ Assuming these data are representative of adult Americans and that an adult American is selected at random, use the given information to estimate the following probabilities. a. \(P(\) tattoo \()\) b. \(P(\) tattoo \(\mid\) age \(18-29)\) c. \(P(\) tattoo \(\mid\) age \(30-50\) ) d. \(P(\) age \(18-29 \mid\) tattoo \()\)

Short Answer

Expert verified
a. \(P(\text{tattoo}) = 0.24\), b. \(P(\text{tattoo | age 18-29}) = 0.36\), c. \(P(\text{tattoo | age 30-50}) = 0.12\), d. \(P(\text{age 18-29 | tattoo}) = 0.75\)

Step by step solution

01

Calculate Total

First, we calculate the total number of people surveyed by summing up all the numbers in our table. In the table, there are 18 + 32 people aged 18-29 and 6 + 44 people aged 30-50 which gives us a total of 100 people.
02

Calculate P(Tattoo)

To find the total probability of having a tattoo, we add the number of people who have tattoos (18 from age group 18-29 and 6 from age group 30-50 to get 24). We then divide this by the total number of people to get a probability of \(\frac{24}{100}=0.24\).
03

Calculate P(Tattoo|Age 18-29)

To find the probability of having a tattoo given the person is between 18-29 years, we consider only people in this age group. The number of people aged between 18-29 with a tattoo is 18, and the total number of people aged between 18-29 is 18 + 32 = 50. Therefore, the probability is \(\frac{18}{50}=0.36\).
04

Calculate P(Tattoo|Age 30-50)

To find the probability of having a tattoo given the person is between 30-50 years, we consider only people in this age group. The number of people aged between 30-50 with a tattoo is 6, and the total number of people aged between 30-50 is 6 + 44 = 50. Therefore, the probability is \(\frac{6}{50}=0.12\).
05

Calculate P(Age 18-29|Tattoo)

To find the probability of being aged between 18-29 given that the person has a tattoo, we consider only people who have a tattoo. The number of people with a tattoo between 18-29 years is 18, and the total number of people with a tattoo is 18 + 6 = 24. Therefore, the probability is \(\frac{18}{24}=0.75\).

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Key Concepts

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

Statistical Probability
Statistical probability is the measure of the likelihood that an event will occur based on statistical data and historical evidence. It is essentially the ratio of the favorable outcomes to the total number of possible outcomes, often expressed as a percentage or a number between 0 and 1. In the context of our exercise, we used a table that presented data on the number of adults with and without tattoos across two different age groups. To estimate the statistical probability of a randomly selected adult having a tattoo, we calculated the total number of people with tattoos and divided by the overall population considered in the survey.

This approach is widely used in various fields to make informed decisions or predictions based on past occurrences. It's essential to note that the accuracy of statistical probability highly depends on the quality and representativeness of the data. If the survey data in our problem truly represents the adult American population, then the calculated probability of \(P(tattoo)\) is a reliable estimate.
Conditional Probability
Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. It helps us to refine our predictions based on new information. The notation \(P(A | B)\) is used to denote the probability of event A occurring given that B has already happened.

In our example, when we calculated \(P(\text{tattoo} | \text{age } 18-29)\), we were looking for the probability of an individual having a tattoo on the condition that they belong to the 18-29 age group. Similarly, the calculation of \(P(\text{age } 18-29 | \text{tattoo})\) gives us the likelihood that a person is aged 18-29 given they have a tattoo. These conditional probabilities are valuable in understanding the demographics of tattoo ownership within given age groups and can be used by marketers, health professionals, and sociologists to tailor interventions or campaigns.
Probability Calculation
Probability calculation involves determining the chance of a particular event happening. It requires counting the number of times an event can occur and the total number of outcomes. The formula commonly used is \(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). One of the best practices for accurate probability calculation is to ensure that categories do not overlap and that all possible outcomes are accounted for.

In the exercise, we performed several probability calculations, each demonstrating a different principle of probability. When calculating the probability of someone having a tattoo in the general adult population, we used the basic probability formula. For conditional probabilities, we modified the denominator to reflect the condition, like focusing on a specific age group only. Each calculation provided different insights, highlighting the importance of knowing which probability type to calculate in a given scenario.

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Most popular questions from this chapter

Two individuals, \(A\) and \(B\), are finalists for a chess championship. They will play a sequence of games, each of which can result in a win for \(A\), a win for \(\mathrm{B}\), or a draw. Suppose that the outcomes of successive games are independent, with \(P(\) A wins game \()=.3\), \(P(\) B wins game \()=.2\), and \(P(\) draw \()=.5\). Each time a player wins a game, he earns 1 point and his opponent earns no points. The first player to win 5 points wins the championship. For the sake of simplicity, assume that the championship will end in a draw if both players obtain 5 points at the same time. a. What is the probability that \(A\) wins the championship in just five games? b. What is the probability that it takes just five games to obtain a champion? c. If a draw earns a half-point for each player, describe how you would perform a simulation to estimate \(\mathrm{P}(\mathrm{A}\) wins the championship). d. If neither player earns any points from a draw, would the simulation in Part (c) take longer to perform? Explain your reasoning.

A library has five copies of a certain textbook on reserve of which two copies ( 1 and 2 ) are first printings and the other three \((3,4\), and 5\()\) are second printings. \(\mathrm{A}\) student examines these books in random order, stopping only when a second printing has been selected. a. Display the possible outcomes in a tree diagram. b. What outcomes are contained in the event \(A\), that exactly one book is examined before the chance experiment terminates? c. What outcomes are contained in the event \(C\), that the chance experiment terminates with the examination of book 5 ?

The manager of a music store has kept records of the number of CDs bought in a single transaction by customers who make a purchase at the store. The accompanying table gives six possible outcomes and the estimated probability associated with each of these outcomes for the chance experiment that consists of observing the number of CDs purchased by the next customer at the store. $$ \begin{aligned} &\begin{array}{l} \text { Number of CDs } \\ \text { purchased } \end{array} & 1 & 2 & 3 & 4 & 5 & 6 \text { or more } \\ &\begin{array}{c} \text { Estimated } \\ \text { probability } \end{array} & .45 & .25 & .10 & .10 & .07 & .03 \end{aligned} $$ a. What is the estimated probability that the next customer purchases three or fewer CDs? b. What is the estimated probability that the next customer purchases at most three CDs? How does this compare to the probability computed in Part (a)? c. What is the estimated probability that the next customer purchases five or more CDs? d. What is the estimated probability that the next customer purchases one or two CDs? e. What is the estimated probability that the next customer purchases more than two CDs? Show two different ways to compute this probability that use the probability rules of this section.

Only \(0.1 \%\) of the individuals in a certain population have a particular disease (an incidence rate of .001). Of th贸se wh貌 have the disease, \(95 \%\) test possitive wh猫n a certain diagnostic test is applied. Of those who do not have the disease, \(90 \%\) test negative when the test is applied. Suppose that an individual from this population is randomly selected and given the test. a. Construct a tree diagram having two first-generation branches, for has disease and doesn't have disease, and two second-generation branches leading out from each of these, for positive test and negative test. Then enter appropriate probabilities on the four branches. b. Use the general multiplication rule to calculate \(P\) (has disease and positive test). c. Calculate \(P\) (positive test). d. Calculate \(P\) (has disease| positive test). Does the result surprise you? Give an intuitive explanation for the size of this probability.

Is ultrasound a reliable method for determining the gender of an unborn baby? The accompanying data on 1000 births are consistent with summary values that appeared in the online version of the Journal of Statistics Education ("New Approaches to Leaming Probability in the First Statistics Course" [2001]). $$ \begin{array}{ccc} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Female } \end{array} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Male } \end{array} \\ \hline \begin{array}{c} \text { Actual Gender Is } \\ \text { Female } \end{array} & 432 & 48 \\ \begin{array}{c} \text { Actual Gender Is } \\ \text { Male } \end{array} & 130 & 390 \\ \hline \end{array} $$ a. Use the given information to estimate the probability that a newborn baby is female, given that the ultrasound predicted the baby would be female. b. Use the given information to estimate the probability that a newborn baby is male, given that the ultrasound predicted the baby would be male. c. Based on your answers to Parts (a) and (b), do you think that a prediction that a baby is male and a prediction that a baby is female are equally reliable? Explain.
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