1
Probability
Probability: the measure of the likelihood of an event.
All probabilities must be 0 < P < 1, with P(E)= 0 meaning that the event cannot happen (impossibility).
P(E) = 1, means that the event will definitely occur (certainty). Probabilities can be expressed as 0, 1,
fractions or decimals.
Theoretical Probability of an event: P(E) =
number of favorable outcomes
total number of possible outcomes
Example: A standard die has six sides with numbers 1-6 on it. The probability of rolling a 5:
P(5) = 1/6,
because there is 1 side with a 5 on it out of 6 possible sides.
The probability of an event not happening [P(not E)] is calculated by 1 – P(E)
Example:
The probability that it will rain is 5/9. The probability that it will not rain is (1 – 5/9) =
4/9.
Using a tree diagram to show a sample space:
Tree Diagram: A visual method of displaying all the outcomes in a sample space by using “branches” that
resemble a tree.
Example: Bobby has 4 shirts (white, blue, red, and yellow). He also has 3 ties (blue, red, and yellow).
Each morning he selects a shirt and a tie at random. Make a tree diagram for all the possible
combinations of shirt and tie. Then use the tree diagram to find the probability of:
1) A blue shirt
2) A red tie
3) Selecting a yellow shirt and yellow tie.
4) Selecting a yellow shirt or a yellow tie.
(Keep in mind the difference between “and” and “or”.)
SHIRTS
TIES
2
Probability
1) A deli offers sandwiches with three choices of bread (white, wheat, or rye). There are also three
choices for meat (ham, roast beef, and turkey), and a choice of mayo or no mayo. Make a tree
diagram that represents all the possible combinations of bread, meat, and mayo. Then use the tree
diagram to answer the following questions.
a. P(white bread and no mayo)
b. P(not wheat)
c. P(roast beef or mayo)
d. P(tuna)
2) The Jeep dealer offers the Grand Cherokee in four colors (black, white, silver, and yellow). There
is a choice of tinted windows or non-tinted windows. There is also a choice between standard
transmission and automatic. Make a tree diagram to represent the sample space of all the possible
combinations. Find:
a. P(black)
b. P(not white)
c. P(yellow or tinted windows)
d. P(silver and tinted windows)
e. P(standard or automatic)
f. P(standard and automatic)
3) At August Madness the cafeteria will be offering a box lunch that consists of a sandwich, bag of
chips, and a drink. There are 4 different sandwiches (cheese, chicken, turkey, and peanut butter).
There are 4 different types of chips(Doritos, pretzels, potato chips, and fritos). There are 3
different drinks (chocolate milk, water, and orange juice). Make a tree diagram to represent the
sample space of all the possible combinations. Find:
a. P(chocolate milk and water)
b. P(peanut butter or fritos)
c. P(turkey or chicken)
d. P (cheese)
e. P(potato chips and water)
4) Janice picks out an SFP dress code approved outfit at random everyday. She picks a blue or gray
skirt. Then she picks a red, white, or navy polo shirt. Janice then picks either navy or gray
stockings. And finally Janice picks either brown or black shoes. Make a tree diagram to represent
the sample space of all the possible combinations. Find:
a. P(gray stockings and white polo) b. P(green skirt)
c. P(black shoes or brown shoes)
d. P(blue skirt or red polo)
e. P(red polo and a white polo)
3
Histograms
1) Jason played in 20 basketball games. The amount of points he scored in each game is listed below.
a) Complete the frequency table and construct a frequency histogram.
b) Which interval contains the median?
c) What is the mode?
15
41
24
31
26
27
26
35
Interval
0-9
10-19
20-29
30-39
49-49
11
14
8
38
Tally
21
25
36
29
Frequency
17
32
41
38
4
2) A class of 24 students took the SAT. Their scores are listed below.
a) Complete the frequency table and construct a frequency histogram.
b) Which interval contains the median?
c) What is the mode?
1120
710
970
770
Interval
700-799
800-899
900-999
1000-1099
1100-1199
1200-1299
1300-1399
1400-1499
1500-1599
1510
1130
990
1120
740
890
860
1130
Tally
1170
910
1310
1080
Frequency
890
1020
1290
1280
1250
1050
1030
1010
5
3) There are 28 students in a science class. Their grades from their most recent exam are listed below.
a) Complete the frequency table and construct a frequency histogram.
b) Which interval contains the median?
c) What is the mode?
91
87
78
76
72
58
92
78
Interval
51-60
61-70
71-80
81-90
91-100
93
84
52
98
Tally
100
78
86
95
Frequency
57
99
94
83
72
93
100
75
85
91
89
85
6
4) There are 36 students in cor 117. The number of times each student was absent is listed below.
a) Complete the frequency table and construct a frequency histogram.
b) Which interval contains the median?
c) What is the mode?
3
0
10
3
13
4
6
5
6
2
0
6
Interval
0-2
3-5
6-8
9-11
12-14
0
7
4
6
7
0
Tally
0
3
0
5
2
0
Frequency
4
1
0
4
5
2
11
4
5
9
3
1
7
5) There are 24 students in a math class. Their grades from their most recent exam are listed below.
a) Complete the frequency table and construct a frequency histogram.
b) Complete the cumulative frequency table and construct a cumulative freq. hist.
c) Which interval contains the median?
d) What is the mode?
e) What interval contains the lower quartile? f) What interval contains the UQ?
92
72
92
82
57
79
87
58
84
72
92
93
78
84
65
86
88
88
92
92
85
89
83
75
Range
Tally
Frequency
Range
51-60
51-60
61-70
51-70
71-80
51-80
81-90
51-90
91-100
51-100
Cumulative
Frequency
8
6) Javier was the starting pitcher in 28 baseball games. The amount of points he scored in each game is
listed below.
a) Complete the frequency table and construct a frequency histogram.
b) Which interval contains the median?
c) What is the mode?
d) Which interval contains the lower quartile?
e) Which contains the UQ?
6
8
2
0
6
7
8
9
7
6
3
6
4
3
5
3
2
1
9
6
2
9
6
5
4
5
6
3
Range
Tally
Frequency
Range
0-1
0-1
2-3
0-3
4-5
0-5
6-7
0-7
8-9
0-9
Cumulative
Frequency