PROBABILITY
BASIC PROBABILITY
1. In drawing a card from a standard deck without looking, find P(any heart).
2. Each of ten pieces of paper contains a different number from {0,1,2,3,4,5,6,7,8,9}. The
pieces of paper are folded and placed into a paper bag. In selecting a piece of paper without
looking, find P(7).
3. An urn contains 5 marbles, all the same size. Two marbles are black and the other three are
white. In selecting a marble without looking, find P(black).
4. Using the lettered tiles from a word game, a boy places 26 tiles face down on a table, one tile
for each letter of the alphabet. After mixing up the tiles, he takes one. What is the
probability that the tile contains one of the letters in the word MATH?
5. Alma is playing a game in which each player must roll a die. To win, Alma must roll a
number greater than 4. What is the probability that Alma will win on her next turn?
6. A fair die is tossed. Find the probability for each of the following events.
a) The number 3 appears.
b) An even number appears.
c) A number less than 3 appears.
d) A number greater than 3 appears.
e) A number greater than or equal to 3 appears.
7. A spinner is divided into 5 equal regions, numbered 1 through 5. An arrow is spun and lands
in one of the regions. For each part of this question state the probability of the event.
a) The number 3 appears.
b) An even number appears.
1
2
c) A number less than 3 appears.
d) An odd number appears.
5
3
e) A number greater than 3 appears.
4
f) A number greater than or equal to 3 appears.
8. A standard deck of 52 cards is shuffled and one card is drawn. What is the probability that
the card is:
a) the queen of hearts?
b) a queen?
c) a heart?
d) a red card?
e) the seven of clubs?
f) a club?
g) an ace?
h) a red seven?
i) a black ten?
j) a picture card?
9. A person does not know the answer to a test question and takes a guess. Find the probability
that answer is correct if the question is:
a) a multiple-choice question with 4 choices
b) a true-false question
c) a question where the choices given are "sometimes, always, or never."
10. A letter is chosen at random from a given word. Find the probability that the letter is a vowel
if the word is:
a) APPLE
b) BANANA
c) GEOMETRY
d) MATHEMATICS
11. A bank contains a nickel, a dime, and a quarter. A person selects one of the coins. What is
the probability that the coin is worth:
a) exactly 10 cents?
b) exactly 3 cents?
c) more than 3 cents?
12. Ted has 2 quarters, 3 dimes, and 1 nickel in his pocket. He pulls out a coin at random. Find
the probability that the coin is worth:
a) exactly 5 cents
b) exactly 10 cents
c) exactly 50 cents
d) less than 25 cents
e) less than 50 cents
f) more than 25 cents
g) more than 1 cent
h) less than 1 cent
13. A single fair die is rolled. Find the probability for each event.
a) The number 8 appears.
b) A whole number appears.
c) The number is less than 5.
d) The number is less than 1.
e) The number is less than 10.
f) The number is negative.
14. A standard deck of 52 cards is shuffled and you pick a card at random. Find the probability
that the card is:
a) a jack
b) a club
c) a star
d) a red club
e) a card from the deck
f) a black club
g) the jack of stars
h) a 17
i) a red 17
15. A girl is holding 5 cards in her hand. They are the 3 of hearts, 3 of diamonds, 3 of clubs, 4
of diamonds, and the 7 of clubs. A player to her left takes one of these cards at random.
Find the probability that the card selected from the 5 cards in the girl's hand is:
a) a 3
b) a diamond
c) a 4
d) a black 4
e) a club
f) 4 of hearts
g) a red card
h) a number card
i) a number less than 8
16. A fair die is rolled once. The sides are numbered 1,2,3,4,5, and 6. Find the probability that
the number rolled is:
a) greater than 2 and odd
b) less than 4 and even
c) greater than 2 and less than 4
d) less than 2 and even
e) less than 6 and odd
f) less than 4 and greater than 3
17. A set of polygons consists of an equilateral triangle, a square, a rhombus, and a rectangle as
shown. One of the polygons is selected at random. Find the probability that the polygon
contains:
a)
b)
c)
d)
e)
all sides congruent and all angles congruent
all sides congruent and all right angles
all sides congruent and two angles not congruent
at least 2 congruent sides and at least 2 congruent angles
at least 3 congruent sides and at least 2 congruent angles
18. A standard deck of 52 cards is shuffled. One card is drawn at random. Find the probability
that the card is:
a) a king or an ace
b) red or an ace
19. A fair die is rolled once. Find the probability of the event described.
a) 4
b) 3 or 4
c) an odd number
d) an odd number or 2
e) less than 4
f) 4 or less
g) 2 or 3 or 4
h) an odd number or 3
i) less than 2 or more than
j) less than 5 or more than 2
20. From a standard deck of cards, one card is drawn. Tell what the probability is that the card
will be:
a) a queen or an ace
b) a queen or a 7
c) a heart or a spade
d) a queen or a spade
e) a queen or a red card
f) jack or queen or king
g) a 7 or a diamond
h) a club or a red card
i) an ace or a picture card
21. A bank contains 2 quarters, 6 dimes, 3 nickels, and 5 pennies. A coin is drawn at random.
Find the probability that the coin is:
a) a quarter
b) a quarter or a dime
c) a dime or a nickel
d) worth 10 cents
e) worth more than 10 cents f) worth 10 cents or less
g) worth 1 cent or more
h) worth more than 1 cent i)a quarter, nickel, or penny
22. The weather bureau predicted 30% chance of rain. Express in fractional form:
a) the probability that it will rain
b) the probability that it will not rain
23. From a standard deck of cards, one card is drawn. Find the probability that the card will be:
a) a club
b) not a club
c) a picture card
d) not a picture card
e) not an 8
f) not a red 6
g) not the queen of spades
24. A letter is chosen at random from the word PROBABILITY. Find the probability that the
letter chosen is:
a) A
b) B
c) C
d) A or B
e) A or I
f) a vowel
g) not a vowel
h) A or B or L
i) A or not A
TREE DIAGRAMS
25. A teacher gives a quiz consisting of two questions. Each question has its answer either true
or false. Using T and F, draw a tree diagram to show all possible ways the questions can be
answered.
26. A quiz consists of three true-false questions. Draw a tree diagram to show all the possible
ways there are to answer the questions on the test.
27. Elizabeth has a number of possible routes to school. She can take either A Street or B Street,
and then turn onto Avenue X, Avenue Y, or Avenue Z. Draw a tree diagram to show all the
possible routes.
28. Draw a tree diagram to show all possibilities, by sex, for a three-child family, then find each
probability.
a) P(all the children are girls)
b) P(all the children are the same sex)
c) P(the two youngest are girls)
d) P(the oldest is a boy)
e) P(the oldest and youngest are the same sex)
f) P(the oldest is a boy and the youngest is a girl)
29. Sheila has to decide what to eat for lunch. She has a choice of 4 salads (caesar, garden, tricolor,
egg), 3 beverages (water, soda, iced tea), and 5 desserts (ice cream, cookies, cake, fruit, donuts).
Draw a tree diagram to all the possible choices Sheila has for lunch.
30. Jack is getting dressed for school. He can't decide what to wear. He has a choice of 4 shirts (red,
purple, gray, black), 2 pants (black ,gray), and 4 pairs of socks (white, black, gray, beige). Draw
a tree diagram to show all the possible outfits Jack can make. Find the probability that his entire
outfit is gray. Find the probability that his outfit consists of nothing that is black.
COUNTING PRINCIPLE
31. There are 10 doors into school and 8 staircases from the first floor to the second. How many
possible ways are there for a student to go from outside the school to a classroom on the second
floor?
32. A tennis club has 15 members, 8 women and 7 men. How many different teams may be formed
consisting of 1 woman and 1 man on each team?
33. A dinner menu lists 2 soups, 7 meats, and 3 desserts. How many different meals consisting of 1
soup, 1 meat, and 1 dessert are possible?
34. Options on a bicycle include 2 types of handlebars, 2 types of seats, and a choice of 15 colors.
The bike may also be ordered in ten-speeds, in three-speeds, or standard. How many possible
versions of a bicycle can customers choose from, if they select the type of handlebars, seat, color,
and speed?
35. A state issues license plates consisting of letters and numbers. There are 26 letters and the letters
may be repeated in a plate; there are 10 digits and the digits may be repeated. Tell how many
possible license plates the state may issue when a license consists of:
a) two letters, followed by three numbers
b) three numbers, followed by three letters
c) four numbers, followed by two letters
36. An ice-cream company offers 31 different flavors. Hilda orders a double scoop cone. In how
many different ways can the clerk put the ice cream on the cone if
a) Hilda wanted two different flavors
b) Hilda wanted the same flavor on both scoops
c) Hilda could not make up her mind and told the clerk, "Anything at all"?
37. Gwen has five skirts and four blouses. How many different skirt-blouse outfits can she choose to
wear on any given day?
38. At the Burger Shoppe, a customer can order a burger rare, medium, or well done. It can be plain,
or have one of these toppings: onions, relish, or mayonnaise. How many different styles of
burgers can be ordered?
39. Four ferryboats make the crossing between point A and point B. How many different ways can a
traveler make a roundtrip?
40. Using the ferryboats in the previous exercise, how many different ways can a traveler make a
roundtrip, but return on a different ferryboat from the one he went on?
PROBABILITY WITH THE COUNTING PRINCIPLE
41. Mr. Gillen may take any of three buses to get to the same train station. The buses are marked A
or B or C. He may then take the 6th Avenue train or the 8th Avenue train to get to work. The
buses and trains arrive at random and are equally likely to arrive. What is the probability that Mr.
Gillen takes the B bus and the 6th Avenue train to get to work?
42. A fair coin and a six-sided die are tossed simultaneously. What is the probability of obtaining:
a) a head on the coin and a 4 on the die?
b) a head and a 3?
c) a tail and a number less than 5?
d) a head and even number?
e) a tail and a number more than 4?
43. Two fair coins are tossed. What is the probability that both land heads up?
44. A fair spinner contains equal regions, numbered 1 through 8. If the arrow is spun twice, find the
probability that:
a) it lands on 7 both times
b) it does not land on 7 either time
45. Jack will fail French with probability 1/3. He will fail chemistry with probability 1/4. What is
the probability he will fail both subjects?
46. Use the information in the previous exercise. What is the probability that Jack will pass French
and fail chemistry?
47. What is the probability that a student will answer two 3-choice multiple choice questions
correctly if the student is forced to guess at both?
48. What is the probability that the student will get the first one right and the second one wrong, using
the information in the previous exercise?
PERMUTATIONS
49. Using the letters E, M, I, T, how many words of four letters can be found if each letter is used
only once in the word?
50. In how many different ways can 5 students be arranged in a row?
51. How many possible three-letter arrangements of the letters X, Y, and Z can be made if each letter
is used only once in each arrangement?
52. How many different 4-digit numbers can be made using the digits 2, 4, 6, and 8 if each digit
appears only once in each number?
53. In a game, Gary held exactly one club, one diamond, one heart, and one spade. In how many
different ways can Gary arrange these four cards in his hand?
54. In how many different ways can 60 people line up to buy tickets at a theater?
55. In how many different ways can the librarian put 35 different novels on a shelf, with one book
following another?
PERMUTATIONS WITH RESTRICTIONS
56. There are 9 players on a baseball team. The manager must establish a batting order for the
players at each game. The pitcher will bat last. How many different batting orders are possible
for the 8 remaining players on the team?
57. Find the number of permutations of the letters A, E, X, Q, B, and R if the first letter must be a A.
58. Find the number of permutations of the letters A, E, X, Q, B, and R if the first and last letters
must be vowels.
59. How many ways can six books be arranged if the first and last must always be the same two
books?
60. How many ways can five persons be seated in a row of five seats if one of them, the guest of
honor, must be in the center?
61. Give the number of permutations of the letters in PARIS for each situation.
a) There are no restrictions.
b) The first letter must be P.
c) The first letter must be a vowel.
d) The first letter cannot be a vowel.
e) The letter R must be in the middle place.
e) The last two letters must be RP in that
order.
62. Give the number of permutations of the letters QUICKLY for each situation.
a) The last letter must be Y.
b) The last letter cannot be Y.
c) The last two letters must be C and K
d) The letter Q must be first and the
in either order.
letter Y must be last.
PERMUTATIONS TAKEN r AT A TIME
63. How many 3-letter words can be formed from the given letters, if each letter is used only once in
a word?
a) LION
b) TIGER
c) MONKEY
d) LEOPARD
e) MAN
64. There are 30 students in a class. Everyday, the teacher calls on different students to write
homework problems on the board, with each problem done only by one student. In how many
ways can the teacher call students to the board if the homework consists of:
a) only one problem?
b) 2 problems?
c) 3 problems?
65. Tell how many possible winning orders there are for a horse race where 3 horses finish in winning
positions and there are:
a) 7 horses
b) 9 horses
c) 11 horses
66. A class has 31 students. They elect 4 people to office, namely the President, Vice-President,
Secretary, and Treasurer. In how many possible ways can 4 people be elected from this class?
67. A lottery ticket contains a four-digit number. How many possible four-digit numbers are there
when:
a) a digit may appear only once in the number?
f) digits may appear more than once in the number?
PERMUTATIONS WITH REPETITIONS
68. Find the number of permutations of the letters in each of the following.
a) BUBBLE
b) GOBBLE
c) POP
d) CLAMMY
e) LOOK
f) HAPPY
g) SORRY
h) WRITTEN
i) TROPICAL
j) MUMMY
k) QUADRATIC
l) BASKETBALL
COMBINATIONS
69. There are 10 teachers in the science department. How many 4-person committees can be formed
in the department if:
a) the committee consists of any four teachers
b) the committee must consist of Mrs. Hill and Mr. Simpson
70. In a class there are 10 boys and 20 girls. Find the number of ways that a team of 3 students can
be selected if the team consists of any 3 students.
71. A coach selects players for a team. If the coach pays no attention to the positions individuals will
play while making this first selection, how many teams may be formed?
a) Of 14 candidates, the coach needs 5 for a basketball team.
c) Of 16 candidates, the coach needs 11 for a football team.
d) Of 13 candidates, the coach needs 9 for a baseball team.
72. A disc jockey has 25 records at hand, but has time to play only 22 on the air. How many groups
of 22 records can be selected?
73. There are 12 roses growing in Rebecca's garden. How many different bunches consisting of 7
roses can be cut and brought into the house?
PERMUTATION OR COMBINATION
74. Decide if each of the following is a combination or permutation situation.
a) selecting three persons to go out and get pizza for everybody
b) selecting three persons to be editor-in-chief, photo editor, and feature editor for the school
yearbook
c) seating six persons in six seats
d) finding the number of orders in which five prize-winning tickets can be pulled from a hat if
all prizes are alike
e) finding the number of orders in which five prize-winning tickets can be pulled from a hat if
the prizes are different
f) dealing five cards to a poker player
g) choosing four kinds of sandwiches to serve at the picnic
h) choosing a committee from the faculty of a school
i) picking the batting order from a list of fifteen players
MEAN, MEDIAN, MODE
75. Find the mean, median, and mode for each of the following sets of numbers.
a) 23, 26, 21, 19, 35, 26
b) 90, 92, 92, 85, 80, 35, 72
c) 3, 6, 7, 4, 12, 4, 7, 16, 12, 13, 5, 19
e) 76, 82, 34, 54, 12, 67, 55, 9, 11, 13, 5
HISTOGRAMS
76. Towering Ted McGurn is the star of the school's basketball team. The number of points scored
by Ted in his last 20 games are:
36, 32, 28, 30, 33, 36, 24, 33, 29, 30,
30, 25, 34, 36, 34, 31, 36, 29, 30, 34
a) Copy and complete the table.
b) Construct a frequency histogram based on the data found in part a.
c) Answer the following questions for this set of data:
1. Which interval contains the greatest frequency?
2. What is the mean of this data to the nearest tenth?
3. Which interval contains the median?
4. Which interval contains the mode?
5. In how many games did Ted score fewer then 26 points?
Interval
35-37
32-34
29-31
26-28
23-25
Tally
Frequency___
77. Thirty students on the track team were timed in the 200-meter dash. Each student's time was
recorded to the nearest tenth of a second. The times were:
29.3, 31.2, 28.5, 37.6, 30.9, 26.0, 32.4, 31.8, 36.6, 35.0,
38.0, 37.0, 22.8, 35.2, 35.8, 37.7, 38.1, 34.0, 34.1, 28.8,
29.6, 26.9, 36.9, 39.6, 29.9, 30.0, 36.1, 38.2, 37.8, 36.0
a) Copy and complete the table.
b) Construct a frequency histogram based on the data found in part a.
c) Which interval contains the median?
Interval
37.0-40.9
33.0-36.9
29.0-32.9
25.0-28.9
21.0-24.9
Tally
Frequency______
78. Thirty-six students' math scores from their last exam are shown below:
56, 82, 76, 75, 87, 86, 68, 76, 74, 94, 96, 86
78, 72, 85, 73, 59, 85, 75, 96, 78, 80, 98, 74
70, 60, 88, 72, 64, 92, 87, 76, 76, 87, 84, 63
a)
b)
c)
d)
Copy and complete the frequency table.
Construct a frequency histogram based on the data found in part a.
Which interval contains the median?
Which interval contains the mode?
Interval
50-59
60-69
70-79
80-89
90-99
Tally
Frequency_______
79. The school is having a magazine sale. Listed below is the number of magazines sold by twentyfive students.
63, 1, 58, 17, 34, 3, 40, 25, 65, 8, 71, 31, 52
68, 79, 5, 41, 32, 62, 4, 47, 43, 75, 71, 7
a) Copy and complete the table.
b) Construct a cumulative frequency histogram based on the data found in part a.
Interval
1-15
16-30
31-45
46-60
61-80
Tally
Frequency
Cumulative Frequency