1. Venn Diagrams & Two-Way Tables
The student is able to (I can):
• Use Venn diagrams and two-way tables to organize data
and calculate probabilities.
2. frequency table – a table that lists a set of values and how
often each one appears
two-way table – a frequency table that displays data
collected from one source that belongs to two
different categories. One category is represented by
rows and the other by columns. It can also be
represented by a Venn diagram.
It can sometimes seem hard to get started when filling out a
two-way table or Venn diagram. The big thing to remember
is that the categories in the diagram or rows and columns in
the table must always add up to the numbers given in the
problem. Always double-check the totals before moving on
to the probability problems.
3. Example: The Venn diagram shows the results of a survey in
which 80 students were asked whether they play a
musical instrument and whether they speak more
than one language.
Survey of 80 Students
Play an
instrument
Speak more
than one
language
25 30
16
9
4. Complete the two-way table using the information in the
Venn diagram.
Plays an
instrument
Does NOT play an
instrument
TOTAL
Speaks more than
one language 16 30
Is NOT
multi-lingual 25 9
TOTAL 80
Survey of 80 Students
Play an
instrument
Speak more
than one
language
25 30
16
9
5. Complete the two-way table using the information in the
Venn diagram.
Plays an
instrument
Does NOT play an
instrument
TOTAL
Speaks more than
one language 16 30 46
Is NOT
multi-lingual 25 9 34
TOTAL 41 39 80
Survey of 80 Students
Play an
instrument
Speak more
than one
language
25 30
16
9
6. Use the table to find the following probabilities:
1. What is the probability that a student chosen at random
speaks more than one language?
Plays an
instrument
Does NOT play an
instrument
TOTAL
Speaks more than
one language 16 30 46
Is NOT
multi-lingual 25 9 34
TOTAL 41 39 80
46 23
0.575
80 40
= =
7. Use the table to find the following probabilities:
2. What is the probability that a student chosen at random
speaks more than one language and does not play an
instrument?
Plays an
instrument
Does NOT play an
instrument
TOTAL
Speaks more than
one language 16 30 46
Is NOT
multi-lingual 25 9 34
TOTAL 41 39 80
30 3
0.375
80 8
= =
8. Use the table to find the following probabilities:
3. What is the probability that a multi-lingual student
chosen at random does not play an instrument?
Plays an
instrument
Does NOT play an
instrument
TOTAL
Speaks more than
one language 16 30 46
Is NOT
multi-lingual 25 9 34
TOTAL 41 39 80
30 15
0.652
46 23
=
9. Example: A teacher wanted to know how many students had
prepared for a recent Geometry test. He surveyed 50
of his students.
Complete the two-way table and then answer the questions.
Studied Did NOT Study TOTAL
Passed the Test 6
Failed the Test 10
TOTAL 38 50
10. Example: A teacher wanted to know how many students had
prepared for a recent Geometry test. He surveyed 50
of his students.
Complete the two-way table and then answer the questions.
Studied Did NOT Study TOTAL
Passed the Test 6 40
Failed the Test 10
TOTAL 38 12 50
11. Example: A teacher wanted to know how many students had
prepared for a recent Geometry test. He surveyed 50
of his students.
Complete the two-way table and then answer the questions.
Studied Did NOT Study TOTAL
Passed the Test 34 6 40
Failed the Test 4 6 10
TOTAL 38 12 50
12. 1. What is the probability that a student selected at random
studied for the test?
2. What is the probability that a random student studied for
the test and failed it?
Studied Did NOT Study TOTAL
Passed the Test 34 6 40
Failed the Test 4 6 10
TOTAL 38 12 50
38 19
0.76
50 25
= =
4 2
0.08
50 25
= =
13. 3. What is the probability that a student who studied also
passed the test?
4. What is the probability that a student who failed the test
also studied?
Studied Did NOT Study TOTAL
Passed the Test 34 6 40
Failed the Test 4 6 10
TOTAL 38 12 50
34 17
0.895
38 19
=
4 2
0.4
10 5
= =
14. Example: A gym has 150 members. 112 of the members use
the gym, and 68 go to the classes. 14 of the
members don’t use the gym or go to classes.
Use this information to complete the Venn diagram
and answer the questions.
Gym Class
15. Example: A gym has 150 members. 112 of the members use
the gym, and 68 go to the classes. 14 of the
members don’t use the gym or go to classes.
Use this information to complete the Venn diagram
and answer the questions.
Gym Class
14
150-14 =136
136-68=68
16. Example: A gym has 150 members. 112 of the members use
the gym, and 68 go to the classes. 14 of the
members don’t use the gym or go to classes.
Use this information to complete the Venn diagram
and answer the questions.
Gym Class
14
68 44
17. Example: A gym has 150 members. 112 of the members use
the gym, and 68 go to the classes. 14 of the
members don’t use the gym or go to classes.
Use this information to complete the Venn diagram
and answer the questions.
Gym Class
14
68 44 24
18. 1. What is the probability that a member chosen at random
uses both the gym and goes to classes?
Gym Class
14
68 44 24
44 22
0.293
150 75
=
19. 2. What is the probability that a member chosen at random
uses the gym only?
Gym Class
14
68 44 24
68 34
0.453
150 75
=
20. 3. What is the probability that a member who uses the gym
also goes to classes?
Gym Class
14
68 44 24
44 11
0.393
112 28
=
