Chapter 8

Use the formula for \(_{n} C_{r}\) to solve To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers ( 1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

Give an example of an event whose probability must be determined empirically rather than theoretically.

Find the middle term in the expansion of $$\left(\frac{3}{x}+\frac{x}{3}\right)^{16}$$.

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$5+7+9+11+\cdots+31$$

Use the formula for \(_{n} C_{r}\) to solve To win in the New York State lottery, one must correctly select 6 numbers from 59 numbers. The order in which the selection is made does not matter. How many different selections are possible?

Write a probability word problem whose answer is one of the following fractions: \(\frac{1}{6}\) or \(\frac{1}{4}\) or \(\frac{1}{3}\)

Find the middle term in the expansion of $$\left(\frac{1}{x}-x^{2}\right)^{12}$$.

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$6+8+10+12+\dots+32$$

Solve by the method of your choice. In a race in which six automobiles are entered and there are no ties, in how many ways can the first four finishers come in?

Explain how to find the probability of an event not occurring. Give an example.

The graph shows that U.S. smokers have a greater probability of suffering from some ailments than the general adult population. Exercises \(57-58\) are based on some of the probabilities expressed as decimals, shown to the right of the bars In each exercise, use a calculator to determine the probability, correct to four decimal places. (Graph cannot copy) If the probability an event will occur is \(p\) and the probability it will not occur is \(q\), then each term in the expansion of \((p+q)^{n}\) represents a probability. The probability that a smoker suffers from depression is 0.28 If five smokers are randomly selected, the probability that three of them will suffer from depression is the third term of the binomial expansion of (Graph cannot copy) What is this probability?

Let $$ \begin{array}{l} \left|a_{n}\right|=-5,10,-20,40, \ldots \\\ \left|b_{n}\right|=10,-5,-20,-35, \ldots \end{array} $$ and $$ \left\\{c_{n}\right\\}=-2,1,-\frac{1}{2}, \frac{1}{4}, \dots $$ Find \(a_{10}+b_{10}\)

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$a+a r+a r^{2}+\dots+a r^{12}$$

Solve by the method of your choice. A book club offers a choice of 8 books from a list of \(40 .\) In how many ways can a member make a selection?

Explain how to find the probability of an event not occurring. Give an example.

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$a+a r+a r^{2}+. . . +a r^{12}$$

Use a system of two equations in two variables, \(a_{1}\) and \(d,\) to solve Exercises \(59-60\) Write a formula for the general term (the \(n\) th term) of the arithmetic sequence whose second term, \(a_{2},\) is 4 and whose sixth term, \(a_{6}\) is 16

Solve by the method of your choice. A medical researcher needs 6 people to test the effectiveness of an experimental drug. If 13 people have volunteered for the test, in how many ways can 6 people be selected?

Explain how to find or probabilities with mutually exclusive events. Give an example.

Explain how to evaluate \(\left(\begin{array}{l}n \\ r\end{array}\right) .\) Provide an example with your explanation.

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$a+(a+d)+(a-2d)$+\cdots+(a+n d)$$

Use a system of two equations in two variables, \(a_{1}\) and \(d,\) to solve Exercises \(59-60\) Write a formula for the general term (the \(n\) th term) of the arithmetic sequence whose third term, \(a_{3},\) is 7 and whose eighth term, \(a_{8},\) is 17

Solve by the method of your choice. Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000,\) second prize is \(\$ 500\) and third prize is \(\$ 100,\) in how many different ways can the prizes be awarded?

Give an example of two events that are not mutually exclusive.

Describe the pattern in the exponents on \(a\) in the expansion of \((a+b)^{n}\).

The bar graphs show changes in educational attainment for Americans ages 25 and older from 1970 to 2007 . Exercises \(61-62\) involve developing arithmetic sequences that model the data. (GRAPH CANT COPY) In \(1970,11.0 \%\) of Americans ages 25 and older had completed four years of college or more. On average, this percentage has increased by approximately 0.5 each year. a. Write a formula for the \(n\) th term of the arithmetic sequence that models the percentage of Americans ages 25 and older who had or will have completed four years of college or more \(n\) years after 1969 b. Use the model from part (a) to project the percentage of Americans ages 25 and older who will have completed four years of college or more by 2019

Solve by the method of your choice. From a club of 20 people, in how many ways can a group of three members be selected to attend a conference?

Explain how to find or probabilities with events that are not mutually exclusive. Give an example.

Describe the pattern in the exponents on \(b\) in the expansion of \((a+b)^{n}\).

The bar graphs show changes in educational attainment for Americans ages 25 and older from 1970 to 2007 . Exercises \(61-62\) involve developing arithmetic sequences that model the data. (GRAPH CANT COPY) In \(1970,55.2 \%\) of Americans ages 25 and older had completed four years of high school or more. On average, this percentage has increased by approximately 0.86 each year. a. Write a formula for the \(n\) th term of the arithmetic sequence that models the percentage of Americans ages 25 and older who had or will have completed four years of high school or more \(n\) years after 1969 . b. Use the model from part (a) to project the percentage of Americans ages 25 and older who will have completed four years of high school or more by 2019 .

Solve by the method of your choice. Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500,\) in how many different ways can the prizes be awarded?

Explain how to find and probabilities with independent events. Give an example.

What is true about the sum of the exponents on \(a\) and \(b\) in any term in the expansion of \((a+b)^{n} ?\).

Company A pays \(\$ 24,000\) yearly with raises of \(\$ 1600\) per year. Company B pays \(\$ 28,000\) yearly with raises of \(\$ 1000\) per year. Which company will pay more in year \(10 ?\) How much more?

Solve by the method of your choice. How many different four-letter passwords can be formed from the letters \(A, B, C, D, E, F,\) and \(G\) if no repetition of letters is allowed?

The president of a large company with \(10,000\) employees is considering mandatory cocaine testing for every employee. The test that would be used is \(90 \%\) accurate, meaning that it will detect \(90 \%\) of the cocaine users who are tested, and that \(90 \%\) of the nonusers will test negative. This also means that the test gives \(10 \%\) false positive. Suppose that \(1 \%\) of the employees actually use cocaine. Find the probability that someone who tests positive for cocaine use is, indeed, a user.

How do you determine how many terms there are in a binomial expansion?

Company A pays \(\$ 23,000\) yearly with raises of \(\$ 1200\) per year. Company B pays \(\$ 26,000\) yearly with raises of \(\$ 800\) per year. Which company will pay more in year \(10 ?\) How much more?

Solve by the method of your choice. Nine comedy acts will perform over two evenings. Five of the acts will perform on the first evening and the order in which the acts perform is important. How many ways can the schedule for the first evening be made?

Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.

Find \(a_{2}\) and \(a_{3}\) for each geometric sequence. $$2, a_{2}, a_{3},-54$$

Solve by the method of your choice. Using 15 flavors of ice cream, how many cones with three different flavors can you create if it is important to you which flavor goes on the top, middle, and bottom?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Assuming the next U.S. president will be a Democrat or a Republican, the probability of a Republican president is 0.5.

Explain how to find a particular term in a binomial expansion without having to write out the entire expansion.

Use the formula for the general term (the nth term) of a geometric sequence to solve suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the fifteenth day of the month?

Solve by the method of your choice. Baskin-Robbins offers 31 different flavors of ice cream. One of its items is a bowl consisting of three scoops of ice cream, each a different flavor. How many such bowls are possible?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The probability that I will go to graduate school is 1.5.

Use the formula for the general term (the nth term) of a geometric sequence to solve suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the thirtieth day of the month?

Exercises \(67-72\) are based on the following jokes about books: \(\cdot\) "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." - Groucho Marx \(\cdot\) "I recently bought a book of free verse. For \(\$ 12\)." \- George Carlin \(\cdot\) "If a word in the dictionary was misspelled, how would we know?" - Steven Wright \(\cdot\) "Encyclopedia is a Latin term. It means 'to paraphrase a term paper." - Greg Ray \(\cdot\) "A bookstore is one of the only pieces of evidence we have that people are still thinking." - Jerry Seinfeld \(\cdot\) "I honestly believe there is absolutely nothing like going to bed with a good book. Or a friend who's read one." \(-\)Phyllis Diller In how many ways can these six jokes be ranked from best to worst?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I toss a coin, the probability of getting heads or tails is \(1,\) but the probability of getting heads and tails is 0.