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Chapter 9: Problem 27
Find the probability for the experiment of tossing a six-sided die twice. The sum is less than 11.
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Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{n=1}^{20}\left(n^{3}-n\right)$$
Expanding an Expression In Exercises \(61-66,\) use the Binomial Theorem to expand and simplify the expression. $$(3 \sqrt{t}+\sqrt[4]{t})^{4}$$
Probability In Exercises \(85-88,\) 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. To find the probability that the sales representative in Exercise 87 makes four sales when the probability of a sale with any one customer is \(\frac{1}{2},\) evaluate the term 8 $$_{4}\left(\frac{1}{2}\right)^{4}\left(\frac{1}{2}\right)^{4}$$ in the expansion of \(\left(\frac{1}{2}+\frac{1}{2}\right)^{8}\)
Arithmetic Mean In Exercises \(101-103,\) use the following definition of the arithmetic mean \(\overline{x}\) of a set of \(n\) measurements \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) $$ \overline{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i} $$ Proof Prove that $$\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}=\sum_{i=1}^{n} x_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} x_{i}\right)^{2}$$
Graphical Reasoning In Exercises 83 and \(84,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. $$f(x)=x^{3}-4 x, \quad g(x)=f(x+4)$$
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