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Chapter 7: Problem 35

A survey in which people were asked how they were planning to prepare their taxes in 2007 revealed the following: \begin{tabular}{lc} \hline Method of Preparation & Percent \\ \hline Computer software & \(33.9\) \\ \hline Accountant & \(23.6\) \\ \hline Tax preparation service & \(17.4\) \\ \hline Spouse, friend, or other & \\ relative will prepare & \(10.8\) \\ \hline By hand & \(14.3\) \\ \hline \end{tabular} What is the probability that a randomly chosen participant in the survey a. Was planning to use an accountant or a tax preparation service to prepare his taxes? b. Was not planning to use computer software to prepare his taxes and was not planning to do his taxes by hand?

Short Answer

Expert verified
#tag_title#Step 2: Calculate the probability of not using computer software and not doing taxes by hand#tag_content#To find the probability that a randomly chosen participant was not planning to use computer software and was not planning to do his taxes by hand, we first add the percentages for "Computer software" and "By hand", and then subtract the result from 100 to get the percentage of people not using either method. Finally, divide by 100 to convert to decimal form. \(P(\text{Not computer software and Not by hand}) = \frac{100 - (33.9 + 14.3)}{100}\) #Answer# a. \(P(\text{Accountant or Tax preparation service}) \approx 0.41\) b. \(P(\text{Not computer software and Not by hand}) \approx 0.518\)

Step by step solution

01

Calculate the probability of using an accountant or tax preparation service

To find the probability that a randomly chosen participant planned to use an accountant or a tax preparation service, we add the percentages for "Accountant" and "Tax preparation service", and then divide by 100 to convert to decimal form. \(P(\text{Accountant or Tax preparation service}) = \frac{23.6 + 17.4}{100}\)

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Key Concepts

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

Finite Mathematics
Finite mathematics is a branch of mathematics that deals with mathematical concepts and techniques that are used in real-world applications and fields such as business, economics, social sciences, and life sciences. This area of mathematics includes topics such as algebra, graph theory, linear programming, set theory, probability, statistics, and finance.

In the context of the exercise, finite mathematics applies to analyzing survey data and making probability calculations. This requires the ability to work with percentages and understand how they represent parts of a whole. When preparing to solve problems like the one given, it's essential to first convert the given percentages into decimal form to facilitate the computation of probabilities, as finite mathematics predominantly operates with numerical and algebraic expressions.
Survey Data Analysis
Survey data analysis involves collecting data from a group of respondents and interpreting that data to answer questions or make decisions. In our exercise, the survey analyzed how individuals planned to prepare their taxes. The first step in analyzing such data is to organize the information, often in the form of a table displaying different categories and the frequency or percentage of responses for each category.

Analyzing survey data isn't just about reading the numbers; it's about understanding the story they tell. In this case, by knowing the probabilities of how people plan to prepare their taxes, one can infer patterns and trends in tax preparation behavior. Accurate analysis can also involve determining the probability of combined categories, such as the likelihood of a respondent using an accountant or a tax preparation service.
Probability Theory
Probability theory is the branch of mathematics that deals with the analysis of random phenomena. The central object of probability theory is a random variable, which is a variable whose possible values are numerical outcomes of a random phenomenon.

In practice, this may involve calculating the likelihood of one or more events happening, such as in our survey example where we calculate the probability of an event 鈥� choosing an accountant or a tax preparation service 鈥� based on given data. Probability is expressed as a number between 0 and 1, with 0 indicating an impossible event and 1 indicating a certain event.

To calculate the probability in part a, as given in the exercise, we add up the chances of each independent event (using an accountant or using a tax preparation service) and express it as a decimal:

Probability Calculations

The calculation would be: \(P(\text{Accountant or Tax preparation service}) = \frac{23.6 + 17.4}{100} = 0.41\) This result means there is a 41% chance that a randomly selected participant used either an accountant or a tax preparation service to prepare their taxes.

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Most popular questions from this chapter

The following table summarizes the results of a poll conducted with 1154 adults. $$\begin{array}{lcccc} \text { Income, \$ } & \text { Range, \% } & \text { Rich, \% } & \text { Class, \% } & \text { Poor, \% } \\ \hline \text { Less than 15,000 } & 11.2 & 0 & 24 & 76 \\ \hline 15,000-29,999 & 18.6 & 3 & 60 & 37 \\ \hline 30,000-49,999 & 24.5 & 0 & 86 & 14 \\ \hline 50,000-74,999 & 21.9 & 2 & 90 & 8 \\ \hline 75,000 \text { and higher } & 23.8 & 5 & 91 & 4 \\ \hline \end{array}$$ a. What is the probability that a respondent chosen at random calls himself or herself middle class? b. If a randomly chosen respondent calls himself or herself middle class, what is the probability that the annual household income of that individual is between $$\$ 30.000$$ and $$\$ 49,999$$, inclusive? c. If a randomly chosen respondent calls himself or herself middle class, what is the probability that the individual's income is either less than or equal to $$\$ 29,999$$ or greater than or equal to $$\$ 50,000$$ ?

A box contains two defective Christmas tree lights that have been inadvertently mixed with cight nondefective lights. If the lights are selected one at a time without replacement and tested until both defective lights are found, what is the probability that both defective lights will be found after exactly three trials?

The estimated probability that a brand-A, a brand-B, and a brand-C plasma TV will last at least \(30,000 \mathrm{hr}\) is \(.90, .85\), and \(.80\), respectively. Of the 4500 plasma TVs that Ace TV sold in a certain year, 1000 were brand A, 1500 were brand \(\mathrm{B}\), and 2000 were brand \(\mathrm{C}\). If a plasma TV set sold by Ace TV that year is selected at random and is still working after \(30,000 \mathrm{hr}\) of use a. What is the probability that it was a brand-A TV? b. What is the probability that it was not a brand-A TV?

Suppose that \(A\) and \(B\) are mutually exclusive events and that \(P(A \cup B) \neq 0\). What is \(P(A \mid A \cup B)\) ?

In a survey conducted in the fall 2006, 800 homeowners were asked about their expectations regarding the value of their home in the next few years; the results of the survey are summarized below: $$\begin{array}{lc} \hline \text { Expectations } & \text { Homeowners } \\ \hline \text { Decrease } & 48 \\ \hline \text { Stay the same } & 152 \\ \hline \text { Increase less than } 5 \% & 232 \\ \hline \text { Increase 5-10\% } & 240 \\ \hline \text { Increase more than 10\% } & 128 \\ \hline \end{array}$$ If a homeowner in the survey is chosen at random, what is the probability that he or she expected his or her home to a. Stay the same or decrease in value in the next few years? b. Increase \(5 \%\) or more in value in the next few years?
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