Transcript Venn Diagrams Lesson 6.2.5 Lesson 6.2.5 Venn Diagrams California Standard: What it means for you: Statistics, Data Analysis and Probability 3.1 Represent all possible outcomes for compound events in an organized.

Venn Diagrams
Lesson 6.2.5
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Lesson
6.2.5
Venn Diagrams
California Standard:
What it means for you:
Statistics, Data Analysis
and Probability 3.1
Represent all possible outcomes
for compound events in an
organized way (e.g., tables, grids,
tree diagrams) and express the
theoretical probability of each
outcome.
You’ll learn about Venn diagrams,
which are useful in helping you to
understand how different events
relate to each other.
Key words:
• Venn Diagram
• outcome
• event
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Lesson
6.2.5
Venn Diagrams
It’s often tricky to figure out in your head how different
events and outcomes are related.
A Venn diagram is a way to show how different events are
related, and they can make probabilities easier to visualize.
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Lesson
6.2.5
Venn Diagrams
A Venn Diagram is a Way to Represent Events
One outcome will often match more than one event.
You can show situations where one or more outcomes
match more than one event using a Venn diagram.
All the possible outcomes are inside the rectangle.
The area where two
circles overlap contains
all the outcomes that
match both events.
Event
A
Event
B
The circles represent
events. All the outcomes
that match an event are
inside that event’s circle.
The next example should make the usefulness of a
Venn diagram clearer.
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Lesson
6.2.5
Example
Venn Diagrams
1
The following are two events for rolling a die once:
Event A: Rolling an even number
Event B: Rolling a number less than 4
Use a Venn diagram to show how many outcomes
match both events.
Solution
The rectangle represents
all possible outcomes.
This means rolling 1, 2, 3, 4, 5, or 6.
The blue circle represents event A, rolling an even number.
The red circle represents event B, rolling less than 4.
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Solution
Solution
continues…
follows…
Lesson
6.2.5
Example
Venn Diagrams
1
The following are two events for rolling a die once:
Event A: Rolling an even number
Event B: Rolling a number less than 4
Use a Venn diagram to show how many outcomes
match both events.
Solution (continued)
The outcome “rolling a 2” is in both
circles. The circles have to overlap,
so that 2 is in both at the same time.
There is 1 outcome that matches both event A and event B.
There is also 1 outcome (5) that matches neither event A
nor event B.
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Lesson
6.2.5
Venn Diagrams
Guided Practice
Andres picks a card from a standard pack.
Event A is “picking a spade.”
Event B is “picking an ace.”
In which section of this Venn diagram do the following
outcomes belong?
1. Ace of clubs
4 – matches event B
2. King of hearts
1
A
B
2 3 4
1 – doesn’t match A or B
3. Ace of spades
3 – matches A and B
4. Three of spades
2 – matches event A
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Solution follows…
Lesson
6.2.5
Venn Diagrams
Guided Practice
5. Sketch a Venn diagram showing the events below
if an integer from 1 to 25 is picked at random.
Place the integers 1 to 25 in the correct areas of the
Venn diagram.
Event A: the number picked is a multiple of 4
Event B: the number picked is a multiple of 6
25 23 22
1 2
A
B
3
4 16 12
6
5
24
8
7
18
20
9 10
14
11 13
21
19
17
15
4, 8, 12, 16, 20, and 24 are multiples of 4,
so they go inside circle A.
6, 12, 18, and 24 are multiples of 6,
so they go inside circle B.
12, and 24 are multiples of 4 and 6,
so they go inside both circles.
All the other values go outside the circles.
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Solution follows…
Lesson
Venn Diagrams
6.2.5
The Circles on a Venn Diagram Don’t Always Overlap
Venn diagrams can show some other situations.
The circles don’t overlap
at all if no outcomes match
both event A and event B.
A
B
A
B
In this diagram, all the
outcomes matching event B
also match event A.
Some outcomes match A, but not B.
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Lesson
6.2.5
Example
Venn Diagrams
2
Draw a Venn diagram showing the following events
for rolling one die:
Event A: Rolling an even number
Event B: Rolling an odd number
Solution
A
B
Outcomes 2, 4, and 6 match event A.
Outcomes 1, 3, and 5 match event B.
No outcomes match both events, so the circles don’t overlap.
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Solution follows…
Lesson
6.2.5
Example
Venn Diagrams
3
Draw a Venn diagram showing the following events
for rolling one die:
Event A: Rolling an odd number
Event B: Rolling less than 6
B
A
Solution
Outcomes 1, 3, and 5 match event A.
Outcomes 1, 2, 3, 4, and 5 match event B.
Outcome 6 does not match either event.
All the outcomes matching event A also match event B.
The circle representing event A is completely inside the
one for event B.
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Solution follows…
Lesson
Venn Diagrams
6.2.5
Guided Practice
Use the Venn diagrams below to answer Exercises 6–8
1.
A
B
2.
3.
A
B
A
4.
B
B
A
Which diagram could show each of the following pairs of
events for picking a number at random from 1 to 100?
6. Event A: odd
Event B: even
2 – a number cannot be odd and even
7. Event A: less than 50
Event B: even
1 – some numbers match only one event and some match both events
8. Event A: less than 50
Event B: less than 20
3 – all numbers that match event B also match event A
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Solution follows…
Lesson
Venn Diagrams
6.2.5
Guided Practice
Use the Venn diagrams below to answer Exercises 9–11
1.
A
B
2.
3.
A
B
A
4.
B
B
A
Which diagram could show each of the following pairs of
events for picking a number at random from 1 to 100?
9. Event A: greater than 39
Event B: less than 74
1 – some numbers match only one event and some match both events
10. Event A: greater than 86 Event B: less than 17
2 – a number cannot be greater than 86 and less than 17 at the same time
11. Event A: a multiple of 6
Event B: a multiple of 3
4 – all multiples of 6 are also multiples of 3
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Solution follows…
Lesson
6.2.5
Venn Diagrams
Independent Practice
This Venn diagram shows two events when a
number from 1 through 20 is chosen at random.
Use it to answer Exercises 1–4.
5
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Find how many outcomes match:
17 14 A
B
19
6
4 20 18 9
1. event A 10
15
16
10
12
1
2. event B 6
3
8 2
13
7
3. both event A and event B 3
4. at least one of events A and B
13
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Solution follows…
Lesson
6.2.5
Venn Diagrams
Independent Practice
This Venn diagram shows two events when a
number from 1 through 20 is chosen at random.
Use it to answer Exercises 5–6.
5
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5. Which of these could be event A: 17
A
B
19
14
6
A. multiple of 4
4 20 18 9
15
B. number less than 16
16
10
12
1
C. even number
3
8 2
13
7
6. Which of these could be event B:
A. multiple of 3
B. number less than 18
C. odd number
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Solution follows…
Lesson
6.2.5
Venn Diagrams
Independent Practice
Sketch Venn diagrams for the following pairs of events.
Place the integers from 1 to 12 in the correct areas of
the Venn Diagram.
9. 10
8.
7.
92
61
E
AC56 7 DB 3
2
1
7. Event A: choosing an odd number
1 98 9 3 4 712
3
7 55 F
Event B: choosing a multiple of 4
11 8
411
10
8
11
2
10
12
4
126
8. Event C: choosing a number less than 6
Event D: choosing a prime number
9. Event E: choosing a number greater than 4
Event F: choosing a multiple of 5
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Solution follows…
Lesson
6.2.5
Venn Diagrams
Round Up
Venn diagrams often don’t give enough information
on their own to figure out probabilities, but they can
still be useful.
In the next Lesson, you’ll see that when you combine
events, a Venn diagram can help you to understand
the situation better.
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