Venn Diagrams Lesson 6.2.5

Venn Diagrams 6.2.5 California Standard: What it means for you: Lesson 6.2.5 Venn Diagrams California Standard: 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. What it means for you: 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

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.

All the possible outcomes are inside the rectangle. 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. Event A Event B The circles represent events. All the outcomes that match an event are inside that event’s circle. The area where two circles overlap contains all the outcomes that match both events. The next example should make the usefulness of a Venn diagram clearer.

Lesson 6.2.5 Venn Diagrams Example 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. Solution continues… Solution follows…

Lesson 6.2.5 Venn Diagrams Example 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.

Venn Diagrams 6.2.5 Guided Practice 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 2. King of hearts 3. Ace of spades 4. Three of spades 1 2 3 4 A B 4 – matches event B 1 – doesn’t match A or B 3 – matches A and B 2 – matches event A Solution follows…

Venn Diagrams 6.2.5 Guided Practice 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 4, 8, 12, 16, 20, and 24 are multiples of 4, so they go inside circle A. A B 19 1 2 3 5 7 9 11 13 14 15 17 21 22 23 25 10 6, 12, 18, and 24 are multiples of 6, so they go inside circle B. 4 8 12 16 20 24 6 18 12, and 24 are multiples of 4 and 6, so they go inside both circles. All the other values go outside the circles. Solution follows…

In this diagram, all the outcomes matching event B also match event A. Lesson 6.2.5 Venn Diagrams The Circles on a Venn Diagram Don’t Always Overlap Venn diagrams can show some other situations. A B The circles don’t overlap at all if no outcomes match both event A and event B. A B In this diagram, all the outcomes matching event B also match event A. Some outcomes match A, but not B.

Lesson 6.2.5 Venn Diagrams Example 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. Solution follows…

Lesson 6.2.5 Venn Diagrams Example 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. Solution follows…

Venn Diagrams 6.2.5 Guided Practice Lesson 6.2.5 Venn Diagrams Guided Practice Use the Venn diagrams below to answer Exercises 6–8 1. A B 2. A B 3. B A 4. A B 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 7. Event A: less than 50 Event B: even 8. Event A: less than 50 Event B: less than 20 2 – a number cannot be odd and even 1 – some numbers match only one event and some match both events 3 – all numbers that match event B also match event A Solution follows…

Venn Diagrams 6.2.5 Guided Practice Lesson 6.2.5 Venn Diagrams Guided Practice Use the Venn diagrams below to answer Exercises 9–11 1. A B 2. A B 3. B A 4. A B 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 10. Event A: greater than 86 Event B: less than 17 11. Event A: a multiple of 6 Event B: a multiple of 3 1 – some numbers match only one event and some match both events 2 – a number cannot be greater than 86 and less than 17 at the same time 4 – all multiples of 6 are also multiples of 3 Solution follows…

Venn Diagrams 6.2.5 Independent Practice 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. Find how many outcomes match: 1. event A 2. event B 3. both event A and event B 4. at least one of events A and B 1 2 3 4 A B 5 6 7 8 9 11 13 12 14 15 17 16 19 18 20 10 10 6 3 13 Solution follows…

Venn Diagrams 6.2.5 Independent Practice 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. Which of these could be event A: A. multiple of 4 B. number less than 16 C. even number 6. Which of these could be event B: A. multiple of 3 B. number less than 18 C. odd number 1 2 3 4 A B 5 6 7 8 9 11 13 12 14 15 17 16 19 18 20 10 Solution follows…

Venn Diagrams 6.2.5 Independent Practice 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. 7. Event A: choosing an odd number Event B: choosing a multiple of 4 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 8. 9. 7. 1 2 3 4 C D 5 6 7 8 9 11 12 10 1 2 3 4 A B 5 6 7 8 9 11 12 10 1 2 3 4 E 6 7 8 9 11 12 F 5 10 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.