Question

Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 15 purchasers. (a) What are the mean value and standard deviation of the number who want a new copy of the book? (b) What is the probability that the number who want new copies is more than two standard deviations away from the mean value? (c) The bookstore has 10 new copies and 10 used copies in stock. If 15 people come in one by one to purchase this text, what is the probability that all 15 will get the type of book they want from current stock? [Hint: Let X = the number who want a new copy. For what values of X will all 15 get what they want?] (d) Suppose that new copies cost $140 and used copies cost $60. Assume the bookstore currently has 50 new copies and 50 used copies. What is the expected value of total revenue from the sale of the next 15 copies purchased? [Hint: Let h(X) = the revenue when X of the 15 purchasers want new copies. Express this as a linear function.]

          Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 15 purchasers.
(a) What are the mean value and standard deviation of the number who want a new copy of the book? 
(b) What is the probability that the number who want new copies is more than two standard deviations away from the mean value? (c) The bookstore has 10 new copies and 10 used copies in stock. If 15 people come in one by one to purchase this text, what is the probability that all 15 will get the type of book they want from current stock? [Hint: Let X = the number who want a new copy. For what values of X will all 15 get what they want?] 
(d) Suppose that new copies cost $140 and used copies cost $60. Assume the bookstore currently has 50 new copies and 50 used copies. What is the expected value of total revenue from the sale of the next 15 copies purchased? [Hint: Let h(X) = the revenue when X of the 15 purchasers want new copies. Express this as a linear function.]

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Added by Emmanuel G.

Probability with Applications in Engineering, Science, and Technology

Matthew A. Carlton • Jay L. Devore 2nd Edition

Chapter 2

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Solved by Expert Umar Sohail Qureshi

10/01/2021

Step 1

- The problem involves a fixed number of trials (15 purchasers) with two possible outcomes (wanting a new copy or a used copy) and a constant probability of success (30% for a new copy). This is a binomial distribution. Show more…

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Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 15 purchasers. (a) What are the mean value and standard deviation of the number who want a new copy of the book? (b) What is the probability that the number who want new copies is more than two standard deviations away from the mean value? (c) The bookstore has 10 new copies and 10 used copies in stock. If 15 people come in one by one to purchase this text, what is the probability that all 15 will get the type of book they want from current stock? [Hint: Let X = the number who want a new copy. For what values of X will all 15 get what they want?] (d) Suppose that new copies cost $140 and used copies cost $60. Assume the bookstore currently has 50 new copies and 50 used copies. What is the expected value of total revenue from the sale of the next 15 copies purchased? [Hint: Let h(X) = the revenue when X of the 15 purchasers want new copies. Express this as a linear function.]

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00:01 Here we are told that 30 % of all students want to buy a new text for a certain course, and we define that as the success.

00:09 And so 70 % of students want a used copy.

00:13 And we have a random selection of 25 purchasers.

00:20 So we have n equals 25.

00:24 We have probability of success is equal to 0 .3.

00:29 So if we define the random variable x as the number of successes, that is the number of students or people who want to buy the new book, it follows a binomial distribution based on a size of 25 and a probability of success of 0 .3.

00:52 Now for part a, we're asked for the mean value and the standard deviation on the number who want the new copy.

00:58 So that's the mean and the standard deviation on the number of successes.

01:13 So for a binomial random variable, the mean is given by n times p.

01:20 Which is 25 times 0 .3, which comes out to 7 .5, and the standard deviation on x is given by, and that comes out to about 2 .29.

02:02 Now, for part b, we're asked what is the probability that the number of successes will be more than two standard deviations away from the mean value.

02:12 So our mean value is 7 .5, and standard deviation is about 2 .3.

02:20 So basically the question is asking, what is the probability that the number of successes is, well, i'll give the range first.

02:34 So 7 .5 is the mean, plus or minus two standard deviations, so 2 times 2 .29.

02:59 So that's 5 .21 or 9 .79.

03:08 So we're really looking for the probability of being more than two standard deviations above 7 .5 or more than two standard deviations below 7 .5.

03:21 So we can say we're looking for the probability that the number of successes is less than or equal to.

03:30 Now successes, it's a discrete distribution so we can only have integer numbers of successes.

03:35 So if we have five or less successes, we are more than two standard deviations below 7 .5.

03:46 But we can also have at least 10 successes, which would put us more than two standard deviations above 7 .5.

03:58 So this can be written as probability of at most five successes, plus 1 minus the probability of at most 9 successes.

04:25 So this is equal to 0 .193 plus 1 minus 0 .93.

04:37 811 and that comes out to 0 .383...

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