### Venn Diagrams

In **probability**, a **Venn diagram** is a graphic organizer that shows a visual representation for all possible **outcomes** of an experiment and the events of the experiment in ovals. Normally, in probability, the Venn diagram will be a box with overlapping ovals inside. Look at the diagram below:

The \begin{align*}S\end{align*} represents all of the possible outcomes of an experiment. It is called the **sample space**. The ovals \begin{align*}A\end{align*} and \begin{align*}B\end{align*} represent the outcomes of the events that occur in the sample space.

#### Drawing a Venn Diagram

1. 2 coins are tossed one after the other. Event \begin{align*}A\end{align*} consists of the outcomes when tossing heads on the first toss. Event \begin{align*}B\end{align*} consists of the outcomes when tossing heads on the second toss. Draw a Venn diagram to represent this example.

We know that:

\begin{align*}S &= \{HH,HT, TH, TT\}\\ A &= \{HH,HT\}\\ B &= \{HH,TH\}\end{align*}

Notice that event \begin{align*}A\end{align*} and event \begin{align*}B\end{align*} share the Heads + Heads outcome and that the sample space contains Tails + Tails, which is neither in event \begin{align*}A\end{align*} nor event \begin{align*}B\end{align*}.

2. Event \begin{align*}A\end{align*} represents randomly choosing a student from \begin{align*}ABC\end{align*} High School who holds a part-time job. Event \begin{align*}B\end{align*} represents randomly choosing a student from \begin{align*}ABC\end{align*} High School who is on the honor roll. Draw a Venn diagram to represent this example.

We know that:

\begin{align*}S =\end{align*} {students in \begin{align*}ABC\end{align*} High School}

\begin{align*}A =\end{align*} {students holding a part-time job}

\begin{align*}B =\end{align*} {students on the honor roll}

Notice that the overlapping oval for \begin{align*}A\end{align*} and \begin{align*}B\end{align*} represents the students who have a part-time job and are on the honor roll. The sample space, \begin{align*}S\end{align*}, outside the ovals represents students neither holding a part-time job nor on the honor roll.

In a Venn diagram, when events \begin{align*}A\end{align*} and \begin{align*}B\end{align*} occur, the symbol used is **\begin{align*}\cap\end{align*}**. Therefore, \begin{align*}A \cap B\end{align*} is the intersection of events \begin{align*}A\end{align*} and \begin{align*}B\end{align*} and can be used to find the probability of both events occurring. If, in a Venn diagram, either \begin{align*}A\end{align*} or \begin{align*}B\end{align*} occurs, the symbol is **\begin{align*}\cup\end{align*}**. This symbol would represent the union of events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, where the outcome would be in either \begin{align*}A\end{align*} or \begin{align*}B\end{align*}.

3. You are asked to roll a die. Event \begin{align*}A\end{align*} is the event of rolling a 1, 2, or a 3. Event \begin{align*}B\end{align*} is the event of rolling a 3, 4, or a 5. Draw a Venn diagram to represent this example. What is \begin{align*}A \cap B\end{align*}? What is \begin{align*}A \cup B\end{align*}?

We know that:

\begin{align*}S &= \{1, 2, 3, 4, 5, 6\}\\ A &= \{1, 2, 3\}\\ B &= \{3, 4, 5\}\end{align*}

\begin{align*}A \cap B &= \{3\}\\ A \cup B &= \{1, 2, 3, 4, 5\}\end{align*}

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### Example

#### Example 1

Let’s say our sample space is the numbers from 1 to 10. Event \begin{align*}A\end{align*} is randomly choosing one of the odd numbers from 1 to 10, and event \begin{align*}B\end{align*} is randomly choosing one of the prime numbers from 1 to 10. Remember that a prime number is a number whose only factors are 1 and itself. Draw a Venn diagram to represent this example. What is \begin{align*}A \cap B\end{align*}? What is \begin{align*}A \cup B\end{align*}?

We know that:

\begin{align*}S &= \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\\ A &= \{1, 3, 5, 7, 9\}\\ B &= \{2, 3, 5, 7\}\end{align*}

Notice that 3 of the prime numbers are part of both sets and are, therefore, in the overlapping part of the Venn diagram. The numbers 4, 6, 8, and 10 are the numbers not part of \begin{align*}A\end{align*} or \begin{align*}B\end{align*}, but they are still members of the sample space.

\begin{align*}A \cap B &= \{3, 5, 7\}\\ A \cup B &= \{1, 2, 3, 5, 7, 9\}\end{align*}

### Review

- \begin{align*}ABC\end{align*} High School is debating whether or not to write a policy where all students must have uniforms and wear them during school hours. In a survey, 45% of the students wanted uniforms, 35% did not, and 10% said they did not mind a uniform and did not care if there was no uniform. Represent this information in a Venn diagram.
- \begin{align*}ABC\end{align*} High School is debating whether or not to write a policy where all students must have uniforms and wear them during school hours. In a survey, 45% of the students wanted uniforms, and 55% did not. Represent this information in a Venn diagram.
- For question 2, calculate the probability that a person selected at random from \begin{align*}ABC\end{align*} High School will want the school to have uniforms or will not want the school to have uniforms.
- Suppose \begin{align*}A=\{5, 6, 8, 10, 12\}\end{align*} and \begin{align*}B=\{8, 9, 12, 13, 14\}\end{align*}. What is \begin{align*}A \cup B\end{align*}?
- Suppose \begin{align*}A=\{1, 7, 13, 17, 21, 25\}\end{align*} and \begin{align*}B=\{7, 14, 21, 28, 35, 42\}\end{align*}. What is \begin{align*}A \cap B\end{align*}?
- In Jason's homeroom class, there are 11 students who have brown eyes, 5 students who are left-handed, and 3 students who have brown eyes and are left-handed. If there are a total of 26 students in Jason's homeroom class, how many of them neither have brown eyes nor are left-handed?
- If event \begin{align*}A\end{align*} is randomly choosing a vowel from the letters of the alphabet, and event \begin{align*}B\end{align*} is randomly choosing a consonant from the letters of the alphabet, do the ovals in the Venn diagram that represents this situation overlap? Explain your answer.
- Use the following Venn diagram to answer the question:

If the 2 ovals in the Venn diagram above represent events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, respectively, what is \begin{align*}A \cup B\end{align*}?

- Use the following Venn diagram to answer the question:

If the 2 ovals in the Venn diagram above represent events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, respectively, what is \begin{align*}A \cap B\end{align*}?

- In the Venn diagram in question 9, what set represents event \begin{align*}A\end{align*}? What set represents event \begin{align*}B\end{align*}?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 1.1.