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Chapter 7: Problem 36

An author has written a book and submitted it to a publisher. The publisher offers to print the book and gives the author the choice between a flat payment of \(\$ 10,000\) and a royalty plan. Under the royalty plan the author would receive \(\$ 1\) for each copy of the book sold. The author thinks that the following table gives the probability distribution of the variable \(x=\) the number of books that will be sold: \(\begin{array}{lrrrr}x & 1000 & 5000 & 10,000 & 20,000 \\ p(x) & .05 & .30 & .40 & .25\end{array}\) Which payment plan should the author choose? Why?

Short Answer

Expert verified
After comparing the expected royalty gain with the flat payment, the author should choose the payment plan which results in more profit. The choice depends on the calculated expected gain from the number of books sold under the royalty plan.

Step by step solution

01

Understand the Probability Distribution

To begin with, we should understand the probability distribution table properly. The entries in column \(x\) represent the number of books likely to be sold and column \(p(x)\) represents the probability associated with each corresponding number of books. For example, the probability of selling 1000 books is 0.05, the probability of selling 5000 books is 0.30 and so forth.
02

Calculate Expected Pay from Royalty Plan

The expected gain from the royalty plan should be calculated. The expected value is found by summing the products of the probability and the respective pay for each event. It is calculated as \(E(x) = \sum x * p(x)\). So, the expected gain will be \(E(x) = (1000*0.05) + (5000*0.30) + (10000*0.40) + (20000*0.25)\) dollars.
03

Compare with Flat Payment

The calculated value of expected royalty gain must be compared with the flat payment offered. If the expected gain from the royalty plan is more than flat payment of \(\$10,000\), then the author should choose the royalty plan, otherwise the author should go for the flat payment.

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Key Concepts

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

Expected Value
The expected value is a key concept in probability and statistics, representing the average outcome of a random event based on probabilities. To find the expected value of a distribution, you multiply each possible outcome by the probability of that outcome, then sum these products. In the author's scenario, this becomes crucial in deciding between different payment plans.
The expected value formula used is:
  • For each possible outcome, multiply the number of books sold by the probability of selling that number.
  • Add all these values together to find the total expected royalty payout.
In our example:\[ E(x) = (1000 \times 0.05) + (5000 \times 0.30) + (10000 \times 0.40) + (20000 \times 0.25) \] This results in the expected royalty revenue, which we can compare to the flat fee offered.
Decision Making
Decision making involves evaluating the expected outcomes and risks of various options to arrive at the most beneficial choice. In the book author's case, decision making centers around choosing between a flat fee and royalties.

To make a decision effectively, the author should:
  • Consider both the expected revenue from royalties and the guaranteed flat payment.
  • Weigh the certainty of the flat fee against the potential gain (or loss) with royalties.
Ultimately, the choice should reflect the author's risk tolerance. If they prefer a guaranteed revenue, the flat fee might be preferable. If they anticipate higher sales and are comfortable with uncertainty, the royalty could lead to higher earnings.
Probability Theory
Probability theory is a branch of mathematics concerned with analyzing random events. It provides tools to quantify the likelihood of various outcomes, an essential step in many decision-making processes. In this exercise, probability theory helps us model possible outcomes of book sales and evaluate their likelihood.
  • The number of books sold is treated as a random variable.
  • The probabilities assigned to these sales figures create a probability distribution.
Understanding these concepts allows the author to quantify potential outcomes and make informed decisions. This systematic approach is applicable in countless real-world problems beyond royalties, such as insurance risk assessment and market analysis.

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Most popular questions from this chapter

Exercise \(7.8\) gave the following probability distribution for \(x=\) the number of courses for which a randomly selected student at a certain university is registered: \(\begin{array}{lrrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ p(x) & .02 & .03 & .09 & .25 & .40 & .16 & .05\end{array}\) It can be easily verified that \(\mu=4.66\) and \(\sigma=1.20\). a. Because \(\mu-\sigma=3.46\), the \(x\) values 1,2, and 3 are more than 1 standard deviation below the mean. What is the probability that \(x\) is more than 1 standard deviation below its mean? b. What \(x\) values are more than 2 standard deviations away from the mean value (either less than \(\mu-2 \sigma\) or greater than \(\mu+2 \sigma) ?\) What is the probability that \(x\) is more than 2 standard deviations away from its mean value?

A box contains four slips of paper marked \(1,2,3\), and \(4 .\) Two slips are selected without replacement. List the possible values for each of the following random variables: a. \(x=\) sum of the two numbers b. \(y=\) difference between the first and second numbers c. \(z=\) number of slips selected that show an even number d. \(w=\) number of slips selected that show a 4

A grocery store has an express line for customers purchasing at most five items. Let \(x\) be the number of items purchased by a randomly selected customer using this line. Give examples of two different assignments of probabilities such that the resulting distributions have the same mean but quite different standard deviations.

A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the variable \(x=\) actual capacity of a randomly selected tank has a distribution that is well approximated by a normal curve with mean \(15.0\) gallons and standard deviation \(0.1\) gallon. a. What is the probability that a randomly selected tank will hold at most \(14.8\) gallons? b. What is the probability that a randomly selected tank will hold between \(14.7\) and \(15.1\) gallons? c. If two such tanks are independently selected, what is the probability that both hold at most 15 gallons?

The size of the left upper chamber of the heart is one measure of cardiovascular health. When the upper left chamber is enlarged, the risk of heart problems is increased. The paper "Left Atrial Size Increases with Body Mass Index in Children" (International Journal of Cardiology [2009]: 1-7) described a study in which the left atrial size was measured for a large number of children age 5 to 15 years. Based on this data, the authors concluded that for healthy children, left atrial diameter was approximately normally distributed with a mean of \(26.4 \mathrm{~mm}\) and a standard deviation of \(4.2 \mathrm{~mm}\). a. Approximately what proportion of healthy children has left atrial diameters less than \(24 \mathrm{~mm}\) ? b. Approximately what proportion of healthy children has left atrial diameters greater than \(32 \mathrm{~mm}\) ? c. Approximately what proportion of healthy children has left atrial diameters between 25 and \(30 \mathrm{~mm}\) ? d. For healthy children, what is the value for which only about \(20 \%\) have a larger left atrial diameter?
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