Consider n independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is p and the probability of a failure is q = 1 - p. In this context, the term $_{n} C_{k} p^{k} q^{n-k}$ in the expansion of $(p+q)^{n}$ gives the probability of k successes in the n trials of the experiment. The probability of a baseball player getting a hit during any given time at bat is $\frac{1}{4}$. To find the probability that the player gets three hits during the next 10 times at bat, evaluate the term $_{10} \mathrm{C}_{3}\left(\frac{1}{4}\right)^{3}\left(\frac{3}{4}\right)^{7}$ in the expansion of $\left(\frac{1}{4}+\frac{3}{4}\right)^{10}$

Why is the concept of a canonical ensemble required?

For the following item, identify the infinitive or infinitive phrase and determine whether it functions as a noun, an adjective, or r an adverb.

As he went out ... to fetch the cows- ...

Use the concept of real options to explain why large restaurant chains often introduce new concept restaurants that have negative NPVs.

Determine a formula for the set of all points $(x, y)$ for which the distance from $(x, y)$ to $(-5,12)$ is 13.

Draw a scatter diagram displaying the data. Let x be the age of a licensed drivers in years. Let y be the percentage of all fatal accidents (for a given age) due to failure to yield the right of way. For example, the first data pair says that 5% of all fatal accidents involving 37-year-olds are dur to failure to yield the right of way. The Wall Street Journal article reported in the following data:

$$\begin{array}{c|cccccc} \hline x & 37 & 47 & 57 & 67 & 77 & 87 \\ \hline y & 5 & 8 & 10 & 16 & 30 & 43 \\ \hline \end{array}$$

Given, $\Sigma x=372,$ $\Sigma y=112,$ $\Sigma x^{2}=24,814,$ $\Sigma y^{2}=3194,$ $\Sigma x y=8254,$ and $r \approx 0.943$.

Why is the x variable sometimes referred to as the predictor variable?

In this problem, find an equation of a sphere that satisfies the given conditions.

Center $(5,2,-2)$; tangent to the y z-plane

How would you prepare the following aqueous solutions?
(b) $445 \mathrm{~g}$ of $13.0$ mass $\% \mathrm{HCl}$ from $34.1$ mass $\% \mathrm{HCl}$

Use the Rational Root Theorem to list all possible rational roots for each equation. Then find any actual rational roots. $10 x^{3}-7 x^{2}+x-10=0$

Use reference triangles in an appropriate quadrant, to find the angles. $\operatorname{arccot}(-1)$

A flow nozzle is to be installed in a $5$ -in Type $K$ copper tube carrying linseed oil at $77^{\circ} \mathrm{F}$. A mercury manometer is to be used to measure the pressure difference across the nozale when the expected range of the flow rate is from $700\ \mathrm{gal} / \mathrm{min}$ to $1000\ \mathrm{gal} / \mathrm{min}$. The manometer scale ranges from 0 to $8.0$ in of mercury. Find an appropriate diameter of the nozzle.

(a) Why does Don Quixote attack the windmills? (b) Connect: How does he explain his failure to conquer them? (c) Hypothesize: What would happen to his dreams of knightly adventure if he admitted the truth about the windmills? Use details from the text to support your answer.

Let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable.. Use a 0.05 significance level. Heights (cm) and weights (kg) are measured for 100 randomly selected adult males. The 100 paired measurements yield $\overline x$ = 173.79 cm, $\overline y$ = 85 .93 kg, r = 0.418, P-value = 0.000, and $\hat y$ = - 106 + 1.10x. Find the best predicted value of y (weight) given an adult male who is 180 cm tall.