Chapter 8

Conditional Probability and Dependent Events Three dice are rolled. If the first die shows a \(4,\) find the probability that the sum of the three dice is less than 12

Determine if \(f\) is an arithmetic sequence. \(f(n)=n^{2}-n+2\)

Evaluate the expression. \(\left(\begin{array}{l}9 \\ 4\end{array}\right)\)

Write out the terms of the series and then evaluate it. $$\sum_{k=2}^{6}(5-2 k)$$

Evaluate the expression. \(\left(\begin{array}{c}20 \\ 18\end{array}\right)\)

Write out the terms of the series and then evaluate it. $$\sum_{k=1}^{7} k^{3}$$

Evaluate the expression. \(\left(\begin{array}{c}100 \\ 2\end{array}\right)\)

Write out the terms of the series and then evaluate it. $$\sum_{k=1}^{4} 5(2)^{k-1}$$

Conditional Probability and Dependent Events Numbers Suppose a number from 1 to 15 is selected at random. Find the probability of each event. A. The number is odd. B. The number is even. C. The number is prime. (Hint: A natural number greater than 1 that has only itself and 1 as factors is called a prime number.) D. The number is prime and odd. E. The number is prime and even.

To win the jackpot in a lottery, one must select 5 different numbers from 1 to \(39 .\) How many ways are there to play this game?

Write out the terms of the series and then evaluate it. $$\sum_{k=4}^{5}\left(k^{2}-k\right)$$

How many ways can a committee of 5 be selected from 8 people?

Writing about Mathematics What values are possible for a probability? Interpret different probabilities and give examples.

Determine if \(f\) is a geometric sequence. $$f(n)=4(2)^{n-1}$$

How many committees of 4 people can be selected from 5 women and 3 men if a committee must have 2 people of each sex on it?

Write the series with summation notation. Let the lower limit equal 1. $$1^{4}+2^{4}+3^{4}+4^{4}+5^{4}+6^{4}$$

Writing about Mathematics Discuss the difference between dependent and independent events. How are their probabilities calculated?

Determine if \(f\) is a geometric sequence. $$f(n)=-3(0.25)^{n}$$

On a test with 6 essay questions, students are asked to answer 4 questions. How many ways can the essay questions be selected? in the second part a student must choose 4 of 5 essay questions. How many ways can the essay questions be selected?

Write the series with summation notation. Let the lower limit equal 1. $$1+\frac{1}{5}+\frac{1}{25}+\frac{1}{125}+\frac{1}{625}$$

Write the series with summation notation. Let the lower limit equal 1. $$1+\frac{4}{3}+\frac{6}{4}+\frac{8}{5}+\frac{10}{6}+\frac{12}{7}+\frac{14}{8}$$

Determine if \(f\) is a geometric sequence. $$f(n)=2(n-1)^{n}$$

How many ways are there to draw a 5-card hand from a 52 -card deck?

Write the series with summation notation. Let the lower limit equal 1. $$2+\frac{5}{8}+\frac{10}{27}+\frac{17}{64}+\frac{26}{125}+\frac{37}{216}$$

How many ways are there to draw 3 red marbles and 2 blue marbles from a jar that contains 10 red marbles and 12 blue marbles?

Write the series with summation notation. Let the lower limit equal 1. $$1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\frac{1}{5^{2}}+\cdots$$

A professor has 3 copies of an algebra book and 4 copies of a calculus text. How many distinguishable ways can the books be placed on a shelf?

Write the series with summation notation. Let the lower limit equal 1. $$1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10,000}+\cdots$$

Determine if \(f\) is a geometric sequence. \(\begin{array}{rrrrrr}n & 1 & 2 & 3 & 4 & 5 \\ f(n) & \frac{1}{2} & \frac{3}{4} & 1 & \frac{5}{4} & \frac{5}{2}\end{array}\)

How many samples of 3 peaches can be drawn from a crate of 24 peaches? (Assume that the peaches are distinguishable.)

A bouquet of flowers contains 3 red roses, 4 yellow roses, and 5 white roses. In how many ways can a person choose 1 flower of each type? (Assume that the flowers are distinguishable.)

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$-5,2,9,16,23,30$$

Permutations Show that \(P(n, n-1)=P(n, n) .\) Give an example that supports your result.

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$5,2,-2,-6,-11$$

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$2,8,32,128,512$$

Explain the difference between a permutation and a combination. Give examples.

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$5.75,5.5,5.25,5,4.75,4.5$$

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$100,110,130,160,200$$

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$0.7,0.21,0.063,0.0189,0.00567$$

(Refer to Example \(12 .\) ) Use properties for summation notation to find the sum. $$\sum_{k=1}^{17}(1-4 k)$$

Suppose that the density of female insects during the first year is 500 per acre with \(r=0.8\). (a) Write a recursive sequence that describes these data, where \(a_{n}\) denotes the female insect density during year \(n\). (b) Find the six terms \(a_{1}, a_{2}, a_{3}, \ldots, a_{6}\). Interpret the results. (c) Find a formula for \(a_{n}\).

Bacteria Growth their size and divide every 40 minutes. (a) Write a recursive sequence that describes this growth where each value of \(n\) represents a 40 -minute interval. Let \(a_{1}=300\) represent the initial number of bacteria per milliter. Find the first five terms. (b) Determine the number of bacteria per milliliter after 10 hours have elapsed. (c) Is this sequence arithmetic or geometric? Explain.

Suppose an insect population density at the beginming of year \(n\) can be modeled by the recursively defined sequence $$ \begin{array}{l}a_{1}=8 \\ a_{n}=2.9 a_{n-1}-0.2 a_{n-1}^{2}, \quad n>1 \\\ \text { (a) Find the population for } n=1,2,3\end{array} $$ (b) Graph the given sequence for \(n=1,2,3, \ldots, 20 .\) Interpret the graph.

If bacteria are cultured in a medium with limited nutrients, competition ensues and growth slows. According to Verhulst's model, the number of bacteria at 40 -minute intervals is given by $$a_{n}=\left(\frac{2}{1+a_{n-1} / K}\right) a_{n-1}$$ where \(K\) is a constant. (a) Let \(a_{1}=200\) and \(K=10,000\). Graph the sequence for \(n=1,2,3, \ldots, 20\). (b) Describe the growth of these bacteria. (c) Trace the graph of the sequence. Make a conjecture as to why \(K\) is called the saturation constant. Test your conjecture by changing the value of \(K\).

The Fibonacci sequence dates back to 1202 . It is one of the most famous sequences in mathematics and can be defined recursively by $$\begin{array}{l}a_{1}=1, a_{2}=1 \\ a_{n}=a_{n-1}+a_{n-2} \quad \text { for } n>2 \\ \text { (a) Find the first } 12 \text { terms of this sequence. } \\\ \text { (b) Compute } \frac{a_{n}}{a_{n-1}} \text { when } n=2,3,4, \ldots, 12 . \text { What hap- } \\ \text { pens to this ratio? }\end{array}$$ (c) Show that for \(n=2,3,\) and 4 the terms of the Fibonacci sequence satisfy the equation $$a_{n-1} \cdot a_{n+1}-a_{n}^{2}=(-1)^{n} \text { . }$$

Verify the formula \(\sum_{k=1}^{n} k=\frac{n(n+1)}{2}\) by using the formula for the sum of the first \(n\) terms of a finite arithmetic sequence.

Suppose an employee's initial salary is 30,000 dollar. (a) If this person receives a 2000 dollar raise for each year of experience, determine a sequence that gives the salary at the beginning of the \(n\) th year. What type of sequence is this? (b) Suppose another employee has the same starting salary and receives a \(5 \%\) raise after each year. Find a sequence that computes the salary at the beginning of the \(n\) th year. What type of sequence is this? (c) Which salary is higher at the beginning of the 10 th year and the 20 th year? (d) Graph both sequences in the same viewing rectangle. Compare the two salaries.

Exercise, Salaries A person has the given starting salary \(S\) and receives a raise \(R\) each year thereafter. (a) Use a formula to calculate the total amount earned over 15 years. (b) Use a calculator to verify this value. $$S=\$ 42,000, R=\$ 1800$$

Exercise, Salaries A person has the given starting salary \(S\) and receives a raise \(R\) each year thereafter. (a) Use a formula to calculate the total amount earned over 15 years. (b) Use a calculator to verify this value. $$S-\$ 35,000, R-\$ 2500$$

The following recursively defined sequence can be used to compute \(\sqrt{k}\) for any positive number \(k .\) \(a_{1}=k ; a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right)\) This sequence was known to Sumerian mathematicians 4000 years ago, but it is still used today. Use this sequence to approximate the given square root by finding a \(6 .\) Compare your result with the actual value. $$\sqrt{11}$$