In each situation, explain whether the selection is made with replacement or without replacement.

a. The three digits in the lottery in the given example.

b. The three students selected to answer questions in the given case study.

c. The five people selected for extra security screening while boarding a particular flight.

d. The two football teams selected to play in the Rose Bowl in a given year.

e. The cars stopped by the police for speeding in five consecutive mornings on the same stretch of highway.

Catrina selects three integers from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and then forms the six possible three-digit integers (leading zero allowed) they determine. For instance, for the selection 1, 3, and 7, she would form the integers 137, 173, 317, 371,713, and 731. Prove that no matter which three integers she initially selects, it is not possible for all six of the resulting three-digit integers to be prime.

The number of carbohydrates (in grams) per 8-ounce serving of yogurt for each of a random selection of brands is listed below. Calculate the true population variance and standard deviation for the number of carbohydrates per 8-ounce serving of yogurt with $95 \%$ confidence. Assume the variable is normally distributed.

$$\begin{array}{lllllllll} 17 & 42 & 41 & 20 & 39 & 41 & 35 & 15 & 43 \\ 25 & 38 & 33 & 42 & 23 & 17 & 25 & 34 & \end{array}$$

Imagine you are trying to test whether a population of flowers is undergoing evolution. You suspect there is selection pressure on the color of the flower: bees seem to cluster around the red flowers more often than the blue flowers. In a separate experiment, you discover blue flower color is dominant to red flower color. In a field, you count 600 blue flowers and 200 red flowers. What would you expect the genetic structure of the flowers to be?

$\partial f / a n$ and $\partial g / \partial n$ denote directional derivatives of scalar fields f and g in the direction of the unit outer normal n to a closed surface S which bounds a solid V of the type described in the divergence theorem. That is, $\partial f / \partial \boldsymbol{n}=\mathbf{V} \mathbf{f} \cdot \boldsymbol{n}$ and $\partial g / \partial \boldsymbol{n}=\nabla g \cdot \boldsymbol{n}$. In each of these exercises prove the given statement. You may assume continuity of all derivatives involved. $\iint_{S} f \frac{\partial g}{\partial n} d S=\iiint_{v} f \nabla^{2} g d x d y d z + \iiint_{V} \nabla f \cdot \nabla g d x d y d z$.

Five candidates {A. B, C, D, and E) have a chance to be selected to be on American Idol. Any subset of them (including none of them or all of them) can be selected, and assume that the selection process is completely random (the subsets of candidates are all equally likely). Find the probability of each of the following events. (a)

$$E_1$$

: "two candidates get selected." (b)

$$E_2$$

: "three candidates get selected." (c)

$$E_3$$

: "three candidates get selected, and A is not one of them."

A restaurant has 8 different appetizers on the menu, as shown in the table. They also offer an appetizer sampler, which contains any 3 of the appetizers served on a single plate. How many different appetizer samplers can be created? The order in which the appetizers are selected does not matter.

(A) Find the number appetizer samplers that are possible if the order of selection does matter . This is the number of permutations of 8 objects taken 3 at a time.

$${ }_8 P_3=\frac{?}{\left({?}-{?}\right)!}=\frac{?}{?}=?$$

(B) Find the number of different ways to select a particular group of appetizers. This is the number of permutations of 3 objects.

$${ }_3 P_3=\frac{?}{\left({?}-{?}\right)!}=\frac{?}{?}=?$$

(C) To find the number of possible appetizer samplers if the order of selection does not matter, divide the answer to part $\mathrm{A}$ by the answer to part $\mathrm{B}$.

So the number of appetizer samplers that can be created is $\frac{?}{?}=?$.

Explain why the answer to Part A was divided by the answer to Part B.

Consider each of these situations. Do you think the proposed sampling method is appropriate? Explain. a) We want to know if there is neighborhood support to turn a vacant lot into a playground. We spend a Saturday afternoon going door to door in the neighborhood, asking people to sign a petition. b) We want to know if students at our college are satisfied with the selection of food available on campus. We go to the largest cafeteria and interview every 10th person in line.

List the key steps necessary to select a ribozyme with ligase activity through Systematic Evolution of Ligands by Exponential Enrichment (SELEX). Assume that you will complete multiple rounds of selection and that you want to diversify the pool each round. You add an RNA molecule that includes a sequence tag to the mixture of your potential ribozymes. If a ribozyme ligates the tag to its own 5' end, the ligating ribozyme can be purified out from the mix.

Look at the following advice from the reading about walking in EL Yunque. After you read the selection, place a (checkmark) next to the sentence that is true.

Consejos para el caminante

• Nunca camine solo. Siempre vaya acompañado.
• Traiga agua y algo para comer.
• Use repelente para insectos.
• No abandone las veredas para no perderse (to get lost).
• No toque (touch) las plantas del bosque.

Use insect repellent._________

A second employee of the computer store in Problem 57 is paid a base salary of $3,000 a month plus a 5% commission on all sales during the month. (A) How much must this employee sell in one month to earn a total of$4,000 for the month? (B) Determine the sales level at which both employees receive the same monthly income.$(C)$ If employees can select either of these payment methods, how would you advise an employee to make this selection?

Write a program that animates the selection-sort algorithm. Create a list that consists of 20 distinct numbers from 1 to 20 in a random order. The elements are displayed in a histogram. Clicking the Step button causes the program to perform an iteration of the outer loop in the algorithm and repaints the histogram for the new list. Color the last bar in the sorted sublist. When the algorithm is finished, display a dialog box to inform the user. Clicking the Reset button creates a new random list for a new start.

A probability experiment consists of selecting a chip at random from a box that contains 40 chips. There are 30 red chips in the box and 10 green chips in the box. Allan plans to carry out a simulation of this probability experiment. He will use a random digit to represent a selection and if the digit is a 1, 2 or 3, the simulated outcome is red. If the random number is not 1, 2 or 3, the simulated outcome will be green. Is Allan’s plan a good one? Explain why or why not.

New Zealand lamb breeders measure the following variance values for their herd.

a. Calculate the broad sense heritability (H$^{2}$) and the narrow sense heritability (h$^{2}$) for each trait in this lamb herd.

b. How would you characterize the potential response to selection ( R) for each trait?