4.1 Counting Rules
Multiplication Rule
1. There are 2 major roads from city X to city Y and 4 major roads from city Y to city Z. How many different trips can be made from city X to city Z passing through city Y?.
2. How many different 3-digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted?
3. A store manager wishes to display 8 brands of shampoo in a row. How many ways can this be done?
Permutation & Combination
4. a) Four tickets are to be selected from 20 tickets. The holder of the first ticket wins a car, the second a motorcycle, the third a bicycle and the fourth a skateboard. In how many different ways can these prizes be awarded? b) Four tickets are to be chosen from 20 tickets. The holders of the winning tickets are to be awarded free trips to the Bahamas. In how many ways can the four winners be chosen?
5. How many different 4-letter words can be formed from the letters in the word
DECAGON?
6. How many different signals can be made by using at least 3 distinct flags if there are 5 different flags from which to select?
7. How many different tests can be made from a test bank of 20 questions if the test consists of
5 questions?
8. An advertising manager decides to have an ad campaign in which 8 special calculators will be hidden at various locations in a shopping mall. If he has 17 locations from which to pick, how many different possibilities can he choose?
9. There are 7 women and 5 men in a department. A committee of 4 members is to be formed from the 12 staff members. i) In how many ways can the committee be formed? ii) In how many ways can this committee be selected if there must be 2 men and 2 women on the committee?

iii) In how many ways can this committee be selected if there must be at least 2 women on the committee?
10. A committee of seven consisting of a chairman, a vice chairman, a secretary, and four other members is to be chosen from a class of 20 students. In how many ways can this committee be chosen?
Distinguishable Permutation
11. In how many ways can the letters of the word DAD be arranged?
12. Find number of different signals that can be formed by displaying 12 flags in a row if 4 flags are green, 5 are red, 2 are blue and 1 is yellow.
13. All the letter of the word PHOTOSYNTHESIS are to be arrange in a row.
(i) Find the number of different words that can be formed?
(ii) How many of the words formed if begin with the letter ‘H’ and end with the letter ‘N’?
Together Problem
14. Four employees at a company picnic are to stand in a row for a group photograph. In how many ways can this be done if
(i) Siti and Ani insist on standing next to each other?.
(ii) Ani and Siti refuse to stand next to each other?.

4.2 Sample Spaces and Probability
1. A die is rolled. List the outcomes for the event of getting an odd number.
2. Roll a die. Find the probability of getting no 9.
3. Roll a die. Find the probability of getting a number less than 7.
4. If two dice are rolled once, find the probability of getting these results:
(a) a sum of 6
(b) doubles
(c) a sum of 7 or 11
(d) a sum greater than 9
(e) a sum less than or equal to 4
5. A couple has three children. Find each probability:
(a) all boys
(b) all girls or all boys
(c) exactly two boys or two girls
(d) at least one child of each gender.
6. Sixty-nine percent of adults favor gun licensing in general. Choose one adult at random. What is the probability that the selected adult does not believe in gun licensing?.
7. In a survey, 16 percent of American children said they use flattery to get their parents to buy them things. If a child is selected at random, find the probability that the child said he or she does not use parental flattery.
8. In a college class of 250 graduating seniors, 50 have jobs waiting, 10 are going to medical school, 20 are going to law school, and 80 are going to various other kinds of graduate schools. Select one graduate at random.
(a) What is the probability that the student is going to graduate school?.
(b) What is the probability that the student is going to medical school?.
(c) What is the probability that the student will have to start paying back his deferred student loans after 6 months (i.e. does not continue in school)?.

4.3 The Addition Rules for Probability
1. At a particular school with 200 male students, 58 play football, 40 play basketball, and 8 play both. What is the probability that a randomly selected male student plays neither sport?.
2. A furniture store decides to select a month for its annual sale. Find the probability that it will be April or May. Assume that all months have equal probability of being selected.
3. An urn contains 6 red balls, 2 green balls, 1 blue ball, and 1 white ball. If a ball is drawn, find the probability of getting a red or a white ball.
4. A grocery store employs cashiers, stock clerks, and deli personnel. The distribution of employees according to marital status is shown here.
Marital Status Cashiers Stock Deli
Married 8
Not Married 5 clerks
12
15 personnel
3
2
If an employee is selected at random, find these probabilities: i) the employee is a stock clerk or married ii) the employee is not married iii) the employee is a cashier or is not married.
5. At a used-book sale, 100 books are adult books and 160 are children’s books. Seventy of the adult books are nonfiction while 60 of the children’s books are nonfiction. If a book is selected at random, find the probability that it is i) fiction ii) not a children’s nonfiction iii) an adult book or a children’s nonfiction.
6. One white die and one black die are rolled. Find the probability that the white die shows a number smaller than 3 or the sum of the dice is greater than 9.
7. A pair of dice is rolled. Event T is defined as the occurrence of a “total of 10 or 11”, and event D is the occurrence of “doubles”. Find P(T or D).

4.4 The Multiplication Rules and Conditional Probability
1. An urn contains 7 blue balls and 3 red balls. A ball is selected and its color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of selecting a blue ball and then a red ball.
2. An urn contains 7 blue balls and 3 red balls. A ball is selected, its color noted and it is not replaced. A second ball is selected and its color noted. Find the probability of selecting a blue ball and then a red ball.
3. In a class containing 12 men and 18 women, 2 students are selected at random to give an impromptu speech. Find the probability that both are women.
4. If 18% of all Americans are underweight, find the probability that if three Americans are selected at random, all will be underweight.
5. At a small college, the probability that a student takes physics and sociology is 0.092. The probability that a student takes sociology is 0.73. find the probability that the student is taking physics, given that he or she is taking sociology.
6. In a pizza restaurant, 95% of the customers order pizza. If 65% of the customers order pizza and a salad, find the probability that a customer who orders pizza will also order a salad.
7. In a statistics class there are 18 juniors and 10 seniors; 6 of the seniors are females, and 12 of the juniors are males. If a student is selected at random, find the probability that the student is a female given that she is a junior.
8. The medal distribution from the 2000 summer Olympic Games is shown in the table:
Gold Silver
United States 39
Russia 32
25
28
China
Australia
Others
28
16
186
Choose one medal winner at random.
16
25
205
Bronze
33
28
15
17
235 i) Given that the winner was from the United States, find the probability that the winner won the gold medal. ii) Find the probability that the winner was from the United States, given that she or he won a gold medal. iii) Are the events ‘medal winner is from United States’ and ‘gold medal was won’ independent? Explain your answer.

9. At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities: i) none have computers ii) at least one has a computer iii) all have computers
10. A lot of portable radios contains 15 good radios and 3 defective ones. If two are selected and tested, find the probability that at least one will be defective.
11. If a die is rolled 3 times, find the probability of getting at least one even number.
12. A manufacturer makes two models of an item: model I, which accounts for 80% of unit sales, and model II, which accounts for 20% of unit sales. Because of defects, the manufacturer has to replace (or exchange) 10% of its model I and 18% of its model II. If a model is selected at random, find the probability that it will be defective.
13. Urn 1 contains 5 red balls and 3 black balls. Urn 2 contains 3 red balls and 1 black ball.
Urn 3 contains 4 red balls and 2 black balls. If an urn is selected at random and a ball is drawn, find the probability it will be red.
14. One white and one black die are rolled. Find the probability that the sum of their numbers is 7 and that the number on the black die is larger than the number on the white die.

4.5 Probability and Counting Rules
1. A toddler has wooden blocks showing the letters C, E, F, H, N, and R. Find the probability that the child arranges the letters in the indicated order: a) In the order FRENCH b) In alphabetical order
2. The president of a large company selects six employees to receive a special bonus. He claims that the six employees are chosen randomly from among the 30 employees, of which 19 are women and 11 are men. What is the probability that no woman is chosen?.
3. An exam has ten true-false questions. A student who has not studied answers all ten questions by just guessing. Find the probability that the student correctly answers the given number of questions: a) All ten questions b) Exactly seven questions
4. To control the quality of their product, the Bright-Light Company inspects three light bulbs out of each batch of ten bulbs manufactured. If a defective bulb is found, the batch is discarded. Suppose a batch contains two defective bulbs. What is the probability that the batch will be discarded?.
5. A monkey is trained to arrange wooden blocks in a straight line. He is then given 11 blocks showing the letters A, B, B, I, I, L, O, P, R, T, Y. What is the probability that the monkey will arrange the blocks to spell the word PROBABILITY?.
6. Find the probability that in a group of eight students at least two people have the same birthday.
7. A student has locked her locker with a combination lock, showing numbers from 1 to 40, but she has forgotten the three-number combination that opens the lock. In order to open the lock, she decides to try all possible combinations. If she can try ten different combinations every minute, what is the probability that she will open the lock within one hour?.
8. A mathematics department consists of ten men and eight women. Six staff members are to be selected at random for the curriculum committee. a) What is the probability that two women and four men are selected?. b) What is the probability that two or fewer women are selected?. c) What is the probability that more than two women are selected?.
9. Twenty students are arranged randomly in a row for a class picture. Paula wants to stand next to Phyllis. Find the probability that she gets her wish.

10. Eight boys and 12 girls are arranged in a row. What is the probability that all the boys will be standing at one end of the row and all the girls at the other end?.
11. A drawer contains an unorganized collection of 50 socks – 20 are red and 30 are blue.
Suppose the lights go out so Kathy can’t distinguish the color of the socks. a) What is the minimum number of socks Kathy must take out of the drawer to be sure of getting a matching pair?. b) If two socks are taken at random from the drawer, what is the probability that they make a matching pair?.
12. Two dice are rolled. Find the probability of each outcome: a) The dice show the same number. b) The dice show different numbers.
13. Find the probability that if 5 different sized washers are arranged in a row, they will be arranged in order of size.
14. License plates are to be issued with 3 letters followed by 4 single digits. How many such license plates are possible? If the plates are issued at random, what is the probability that the license plate says WMQ followed by a number which is divisible by 5?.

Counting Rules (exercises)
1. A company has 2844 employees. Each employee is to be given an ID number that consists of one letter followed by two digits. Is it possible to give each employee a different ID number using this scheme?.
2. In how many ways can five students be seated in a row of five chairs if Jack insists on sitting in the first chair?.
3. Standard automobile license plates in California display a nonzero digit, followed by three letters, followed by three digits. How many different standard plates are possible in this system?.
4. In how many ways can three pizza toppings be chosen from 12 available toppings?.
6. In how many ways can seven students from a class of 30 be chosen for a field trip if Jack must go on the field trip?.
7. In how many ways can a president, vice president, and secretary be chosen from a class of
20 females and 30 males if the president must be a female and the vice president a male?.
8. A group of 22 aspiring thespians contains ten men and twelve women. For the next play the director wants to choose a leading man, a leading lady, a supporting male role, a supporting female role, and eight extras that consists of three women and five men. In how many ways can the cast be chosen?.
9. Five letter “ words” are formed using the letters A, B, C, D, E, F, G. How many such words are possible for each of the following conditions? i) no condition is imposed ii) no letter can be repeated in a word iii) each word must begin with the letter A iv) the letter C must be in the middle v) the middle letter must be a vowel

10. From a group of 30 contestants, six are to be chosen as semifinalists, then two of those are chosen as finalists, and then the top prize is awarded to one of the finalists. In how many ways can these choices be made in sequence?.
11. Three-digit numbers are formed using the digits 2 , 4, 5, and 7, with repetition of digits allowed. How many such numbers can be formed if i) the numbers are less than 700. ii) the numbers are even. iii) the numbers are divisible by 5.
13. A five-person committee consisting of students and teachers is being formed to study the issue of student parking privileges. Of those who have expressed an interest in serving on the committee, 12 are teachers and 14 are students. In how many ways can the committee be formed if at least one student and one teacher must be included?.
14. In how many different ways can the letters of the word ELEEMOSYNARY be arranged?.
15. In how many different ways can five red balls, two white balls, and seven blue balls be arranged in a row?.
16. In how many ways can four men and four women be seated in a row of eight seats if i) the women are to be seated together. ii) the men and women are to be seated alternately by gender.
17. (a) In how many ways can ten students be arranged in a row for a class picture if John and
Jane want to stand next to each other, and Mike and Molly also insist on standing next to each other?.
(b) In how many ways can the ten students be arranged if Mike and Molly insist on standing together but John and Jane refuse to stand next to each other?.
18. In how many ways can a committee of four be chosen from a group of ten if two people refuse to serve together on the same committee?.