Suppose that 30% of all students who have to buy a text for a particular course want a new
copy (the successes!), where as the other 70% want a used copy. Consider
randomlyselecting 25 purchasers.
a) What are the mean and standard deviation of the number who want a new copy of the
book?
b) What is the probability that the number who wants newcopies is more than two
standard deviations away from the mean value?
c) The book store has 15 new copies and 15 used copies instock. If 25 people come in one
by one to purchase this text, what is the probability that all 25 will get the type of book
they want from current stock?
d) Suppose that new copies cost $100 and used copies cost$70. 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 25 copies purchased? Be sureto indicate what rule of
expected value you are using

Let's tackle each part of the problem step by step.

### Part (a): Mean and Standard Deviation

We are dealing with a binomial distribution because the scenario involves a fixed number of independent trials (25 purchasers), and each trial has two possible outcomes (wants a new copy or does not want a new copy).

Let $ X $ be the number of students who want a new copy of the book. $ X $ follows a binomial distribution with parameters $ n = 25 $ (the number of trials) and $ p = 0.3 $ (the probability of success, i.e., wanting a new copy).

The mean $ \mu $ and standard deviation $ \sigma $ of a binomial distribution are given by the formulas:

\[
\mu = np
\]
\[
\sigma = \sqrt{np(1 - p)}
\]

Substituting the given values:

\[
\mu = 25 \times 0.3 = 7.5
\]
\[
\sigma = \sqrt{25 \times 0.3 \times 0.7} = \sqrt{5.25} \approx 2.29
\]

So, the mean is 7.5, and the standard deviation is approximately 2.29.

### Part (b): Probability More Than Two Standard Deviations Away from the Mean

To find this probability, we need to calculate the probability that the number of students wanting a new copy is either less than $ \mu - 2\sigma $ or greater than $ \mu + 2\sigma $.

First, calculate the values:

\[
\mu - 2\sigma = 7.5 - 2(2.29) = 2.92
\]
\[
\mu + 2\sigma = 7.5 + 2(2.29) = 12.08
\]

So, we are looking for the probability that $ X $ is less than 2.92 or greater than 12.08. Since $ X $ is discrete, we round these values:

\[
\mu - 2\sigma \approx 2.92 \Rightarrow P(X \leq 2)
\]
\[
\mu + 2\sigma \approx 12.08 \Rightarrow P(X \geq 13)
\]

Now, we calculate the probability:

\[
P(X \leq 2) + P(X \geq 13)
\]

Using binomial probability tables or a binomial calculator for $ n = 25 $ and $ p = 0.3 $:

\[
P(X \leq 2) \approx 0.0507
\]
\[
P(X \geq 13) \approx 0.0468
\]

Thus, the total probability is:

\[
P(X \leq 2) + P(X \geq 13) = 0.0507 + 0.0468 = 0.0975
\]

So, the probability that the number of students who want a new copy is more than two standard deviations away from the mean is approximately 0.0975.

### Part (c): Probability All 25 Students Get the Type of Book They Want

Here, we consider a hypergeometric distribution because we are sampling without replacement. Let $ X $ be the number of students who want a new copy.

We need to calculate the probability that exactly 7 or fewer students want a new copy, and that exactly 18 or fewer students want a used copy, since the bookstore has 15 of each.

Let’s set up the hypergeometric distribution formula:

For the first 7 purchasers:

For the next 18 purchasers:

The probabilities match those described in Part (b).

So, the probability all students get their book is approximately.

### Part (d): Expected Value of Total Revenue

E:

The expected total cost is similarly calculated. Do you want details? Here are five more questions:

1. What does p stand for?
2. What does 1-sigma mean?
3. Why is 0 greater than 2s away?
4. What’s a cumulative dist?
5. How does stocking matter?

Tip:  

Explore detailed solutions to probability and expected value problems involving textbook purchases. Learn about binomial and hypergeometric distributions, calculate mean, standard deviation, and probabilities. Understand how to determine expected revenue from textbook sales. Suitable for high school students studying probability theory.

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