## Probability with Venn diagrams

#### What is a Venn diagram?

We define Venn diagram as a graphic representation of a group of sets of items (or possible outcomes) where the way each set is related to the others in the diagram can be clearly seen. The sets are always presented as either circles or closed curves, and they overlap with one another when their characteristics are shared, or when the items in one set belong to another too.

Confusing? Just take a look at the Venn diagram example below:

In this 2 circle Venn diagram we can see the composition of the American continent, and divide it between North America and Latin America. Notice that there is a piece of North America which is also part of Latin America, and that is Mexico! (Yes, Mexico is part of North America). Therefore, you can see how a Venn diagram shows the relationship between sets (in this case the two sets are North and Latin America) by depicting them as circles, and whenever an item from a set belongs to other, these circles will overlap.

Venn Diagrams can be a useful tool to represent and solve probability questions. In order to understand what a Venn diagram is and represents in a clearer manner, we have to remember the following details:

**A sample space: All possible outcomes of an experiment**

Therefore, the sample space is the complete box surrounding the circles (or closed curves) that represent the sets.

**Union ($\, \cup$, "or"): $A \, \cup \, B \,$ is the event that either A occurs or B occurs or they both occur.**

**Intersection ($\, \cap$, "and"): $A \, \cup \, B\,$ is the event that both A occurs and B occurs**

**Addition Rule:**

A shortcut formula to finding is:

More on the addition rule will be said in the next lessons. And the basic process will be explained on the Venn diagram problems in the last section of this lesson.

**Complement ($A^{c}$):**

#### Venn diagram problems

Using the Venn diagram definition from our first section, resolve the next example problems.__Example 1__

Looking at Venn diagrams for the representation of events, check out the next example problem: Out of a school of 100 students the number of students enrolled in PE and Band class is given below:
So there are 35 students who are taking only PE classes, 30 students who are taking only Band classes, and 20 students who are taking both PE and Band classes. There are also 15 students who are not enrolled in PE or Band classes.

**1. $\quad$ Replicate this Venn diagram in terms of probabilities of picking a random student from any of these classes.**

In this case we have the next classes:

- PE (Physical education) class = 35 students.
- Band class = 30 students.
- Both PE and Band = 20 students.
- Not enrolled = 15 students.

Therefore, we have a total of: 35 + 30 + 20 + 15 = 100 students, as mentioned in the problem. This total of students is the complete sample space; and so, the probabilities for each class are:

- P(PE) = 35/100
- P(Band) = 30/100
- P(Both PE and Band) = 20/100
- P(Not enrolled) = 15/100

Meaning that each class has a certain amount of students out of 100, and thus the Venn diagram probability depiction goes as:

Where all of the sample space is represented as everything inside the box on the diagram, and has a probability of 1. And each of the classes we have listed, has its probability marked in each of their corresponding areas.

**2. $\quad$ What is the probability of picking a student who is in PE class?**

Therefore, our total probability of picking a student who is enrolled in PE class is equal to 55/100 or 0.55.

__Example 2__

On this problem we take a look at how to use unions and intersections in Venn diagrams.
The following dots represent students who attend each after school activity:

**1. $\quad$ How many students attend only Band Practice, what is the probability of picking a student who attends only Band Practice?**

Therefore, since there are only 7 dots in the area belonging to Band practice ONLY, we have that there are 7 students out of a total of 22, who attend Band practice as their only activity in the afternoons, after school. And so, the probability of picking a student who attends only band practice in the afternoons is equal to: $P(B)$ = 7/22 = 0.3181818.

Such probability belongs to this area on the 3 circle Venn diagram:

**2. $\quad$ How many students attend Soccer Practice AND Band Practice, what is the probability of picking these students?**

Counting the amount of students belonging to the $B \, \cap \, S$ (intersection of Band and Soccer) region as marked above, we have that there are a total of 5 students who attend both band and soccer practice after school, and so, the probability of picking one of these students out of the total of 22 is equal to $P(B \, \cap \, S)$ = 5/22 = 0.227272.

**3. $\quad$ How many students attend Soccer Practice AND Band Practice AND Track Practice, what is the probability of picking these students?**

Therefore, we can clearly see that the amount of students in the intersection $B \, \cap \, S \, \cap \, T$ is equal to 2. And the probability of picking one of these 2 students out of the total of 22 is equal to: $P(B \, \cap \, S \, \cap \, T)$ = 2/22 = 0.09091.

**4. $\quad$ How many students attend Track Practice OR Soccer Practice, what is the probability of picking these students?**

And so, counting the dots in the region for the union of track and soccer practices, we have that there are a total of 15 students who attend track and soccer. Thus, the probability of picking one of these students is equal to: $P(T \, \cup \, S)$ = 15/22 = 0.681818

**5. $\quad$ How many students attend Band Practice OR Soccer Practice, but not Track Practice? What is the probability of picking these students?**

Counting the dots in the region for the union of band and soccer practices, we have that there are a total of 17 students who attend track and soccer. Thus, the probability of picking one of these students is equal to: $P(T \, \cup \, S)$ = 17/22 = 0.772727

__Example 3__

This next problem is dedicated to the addition rule, for extended details on the addition rule take a look at our next lesson on addition rule for OR.
From a deck of cards, 10 cards are drawn at random. From these 10 cards the probability of picking a king is 0.3 and the probability of picking a heart is 0.5. The probability of picking a card that is both a king and a heart (the king of hearts) is 0.1.

**1. $\quad$ Represent these probabilities as a Venn Diagram**

With that in mind, we can conclude that our sample of 10 cards contains: 2 kings and 4 cards that have the heart symbol different from the king of hearts, the king of hearts, and three cards that are left which can be anything but a king or a heart.

And so, we make a Venn diagram from such a sample, and the result is shown below:

Notice that the circle representing the hearts is bigger than that representing the kings, that is because we have more hearts than kings. Also notice that the sample space contains 3 other cards which are not hearts or kings.

**2. $\quad$ What is the probability of picking a king that isnt a heart?**

*King other than king of hearts*)= 2/10 = 0.2.

**3. $\quad$ What is the probability of picking a heart that isnt a king?**

*Heart card other than king of hearts*) = 4/10 = 0.4.

Notice that the probabilities in the Venn diagram are not the same as the ones answering questions 2 and 3; this is because the probabilities in the Venn diagram are for everything that is inside either the black circle or the red circle.

- P(k)=0.3 for everything inside the black circle.
- P(h)=0.5 for everything inside the red circle.

**4. $\quad$ What is the probability of picking a king OR a heart?**

__Example 4__

The probabilities of certain events are given in the Venn Diagram below:
**1. $\quad$ What is the probability of A occurring?**

**2. $\quad$ What is the probability of A NOT occurring?**

**3. $\quad$ What is the probability of $A$ occurring or $B$ occurring?**

**4. $\quad$ What is the probability of neither $A$ nor $B$ occurring?**

This is the end of our lesson. Before we pass onto the next lesson we recommend you to take a look at this handout of probability problems, where you will find some useful Venn diagram examples that can be useful for you to practice even more.

We hope you have enjoyed the lesson of today, see you on the next!

Venn Diagrams can be a useful tool to represent and solve probability questions

Sample space: All possible outcomes of an experiment

Union($\cup$, "or"): A$\cup$B is the event that either A occurs or B occurs or they both occur

Intersection ($\cap$, "and"): A$\cap$B is the event that both A occurs and B occurs

Addition Rule: A shortcut formula to finding is: A$\cup$B = A + B – A$\cap$B

Complement $(A^c)$: All outcomes EXCEPT the event $P(A^c )=1-P(A)$

Sample space: All possible outcomes of an experiment

Union($\cup$, "or"): A$\cup$B is the event that either A occurs or B occurs or they both occur

Intersection ($\cap$, "and"): A$\cap$B is the event that both A occurs and B occurs

Addition Rule: A shortcut formula to finding is: A$\cup$B = A + B – A$\cap$B

Complement $(A^c)$: All outcomes EXCEPT the event $P(A^c )=1-P(A)$