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Connexions module: m16835
1
Probability Topics: Contingency
Tables
∗
Susan Dean
Barbara Illowsky, Ph.D.
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
Abstract
This module introduces the contingency table as a way of determining conditional probabilities.
A contingency table provides a dierent way of calculating probabilities. The table helps in determining
conditional probabilities quite easily. The table displays sample values in relation to two dierent variables
that may be dependent or contingent on one another. Later on, we will use contingency tables again, but in
another manner.
Example 1
Suppose a study of speeding violations and drivers who use car phones produced the following
ctional data:
Speeding violation in No speeding viola- Total
the last year
tion in the last year
Car phone user
Not a car phone user
Total
25
45
70
280
405
685
305
450
755
Table 1
The total number of people in the sample is 755. The row totals are 305 and 450. The column
totals are 70 and 685. Notice that 305 + 450 = 755 and 70 + 685 = 755.
Calculate the following probabilities using the table
Problem 1
P(person is a car phone user) =
Solution
number of car phone users
total number in study
∗ Version
=
305
755
1.11: Feb 12, 2011 11:09 am US/Central
† http://creativecommons.org/licenses/by/3.0/
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
http://cnx.org/content/m16835/1.11/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
Page 1 of 5
Connexions module: m16835
2
Problem 2
P(person had no violation in the last year) =
Solution
number that had no violation
total number in study
=
685
755
Problem 3
P(person had no violation in the last year AND was a car phone user) =
Solution
280
755
Problem 4
P(person is a car phone user OR person had no violation in the last year) =
Solution
305
755
+
685
755
−
280
755
=
710
755
Problem 5
P(person is a car phone user GIVEN person had a violation in the last year) =
Solution
(The sample space is reduced to the number of persons who had a violation.)
25
70
Problem 6
P(person had no violation last year GIVEN person was not a car phone user) =
Solution
405
450
(The sample space is reduced to the number of persons who were not car phone users.)
Example 2
The following table shows a random sample of 100 hikers and the areas of hiking preferred:
Hiking Area Preference
Sex
The Coastline Near Lakes and Streams On Mountain Peaks Total
Female
Male
Total
18
___
___
16
___
41
___
14
___
45
55
___
Table 2
Problem 1
(Solution on p. 5.)
Problem 2
(Solution on p. 5.)
Complete the table.
Are the events "being female" and "preferring the coastline" independent events?
Let F = being female and let C = preferring the coastline.
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
http://cnx.org/content/m16835/1.11/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
Page 2 of 5
Connexions module: m16835
3
a. P(F AND C) =
b. P (F) · P (C) =
Are these two numbers the same? If they are, then F and C are independent. If they are not, then
F and C are not independent.
Problem 3
(Solution on p. 5.)
Find the probability that a person is male given that the person prefers hiking near lakes and
streams. Let M = being male and let L = prefers hiking near lakes and streams.
a. What word tells you this is a conditional?
b. Fill in the blanks and calculate the probability: P(___|___) = ___.
c. Is the sample space for this problem all 100 hikers? If not, what is it?
Problem 4
(Solution on p. 5.)
Find the probability that a person is female or prefers hiking on mountain peaks. Let F = being
female and let P = prefers mountain peaks.
a. P(F) =
b. P(P) =
c. P(F AND P) =
d. Therefore, P(F OR P)
=
Example 3
Muddy Mouse lives in a cage with 3 doors. If Muddy goes out the rst door, the probability that
he gets caught by Alissa the cat is 15 and the probability he is not caught is 54 . If he goes out the
second door, the probability he gets caught by Alissa is 14 and the probability he is not caught is 34 .
The probability that Alissa catches Muddy coming out of the third door is 12 and the probability
she does not catch Muddy is 21 . It is equally likely that Muddy will choose any of the three doors
so the probability of choosing each door is 13 .
Door Choice
Caught or Not Door One Door Two Door Three Total
Caught
Not Caught
Total
1
15
4
15
1
12
3
12
1
6
1
6
____
____
____
____
____
1
Table 3
1
• The rst entry 15
= 15 13 is P(Door One AND Caught).
4
• The entry 15
= 54 13 is P(Door One AND Not Caught).
Verify the remaining entries.
Problem 1
(Solution on p. 5.)
Complete the probability contingency table. Calculate the entries for the totals. Verify that the
lower-right corner entry is 1.
Problem 2
What is the probability that Alissa does not catch Muddy?
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
http://cnx.org/content/m16835/1.11/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
Page 3 of 5
Connexions module: m16835
4
Solution
41
60
Problem 3
What is the probability that Muddy chooses Door One OR Door Two given that Muddy is caught
by Alissa?
Solution
9
19
You could also do this problem by using a probability tree. See the Tree Diagrams (Optional) section of this chapter for examples.
note:
1
1 "Probability Topics: Tree Diagrams (optional)" <http://cnx.org/content/m16846/latest/>
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
http://cnx.org/content/m16835/1.11/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
Page 4 of 5
Connexions module: m16835
5
Solutions to Exercises in this Module
Solution to Example 2, Problem 1 (p. 2)
Hiking Area Preference
Sex
The Coastline Near Lakes and Streams On Mountain Peaks
Female 18
16
11
Male
16
25
14
Total
34
41
25
Total
45
55
100
Table 4
Solution to Example 2, Problem 2 (p. 2)
a. P(F AND C) =
b. P (F) · P (C) =
18
100
45
100
= 0.18
34
· 100
= 0.45 · 0.34 = 0.153
P(F AND C) 6= P (F) · P (C), so the events F and C are not independent.
Solution to Example 2, Problem 3 (p. 3)
a. The word 'given' tells you that this is a conditional.
25
b. P(M|L) = 41
c. No, the sample space for this problem is 41.
Solution to Example 2, Problem 4 (p. 3)
45
a. P(F) = 100
25
b. P(P) = 100
11
c. P(F AND P) = 100
45
d. P(F OR P) = 100 +
25
100
−
11
100
=
59
100
Solution to Example 3, Problem 1 (p. 3)
Door Choice
Caught or Not Door One Door Two Door Three Total
Caught
Not Caught
Total
1
15
4
15
5
15
1
12
3
12
4
12
1
6
1
6
2
6
19
60
41
60
1
Table 5
Glossary
Denition 1: Contingency Table
The method of displaying a frequency distribution as a table with rows and columns to show how
two variables may be dependent (contingent) upon each other. The table provides an easy way to
calculate conditional probabilities.
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
http://cnx.org/content/m16835/1.11/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
Page 5 of 5