Suppose you have the set of integers from 1 to 19. How would you represent the set of odd numbers within that range and the set of prime numbers within that range? Can you think of an easy way to represent the union and intersection of those two different sets of numbers?

### Watch This

James Sousa Set Operations and Venn Diagrams - Part 2 of 2

### Guidance

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*}**sample space**. The ovals \begin{align*}A\end{align*}

#### Example A

2 coins are tossed one after the other. Event \begin{align*}A\end{align*}

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*}

#### Example B

Event \begin{align*}A\end{align*}

We know that:

\begin{align*}S =\end{align*}

\begin{align*}A =\end{align*}

\begin{align*}B =\end{align*}

Notice that the overlapping oval for \begin{align*}A\end{align*}

In a Venn diagram, when events \begin{align*}A\end{align*}**\begin{align*}\cap\end{align*} ∩**. Therefore, \begin{align*}A \cap B\end{align*}

**\begin{align*}\cup\end{align*}**∪ . This symbol would represent the union of events \begin{align*}A\end{align*}

#### Example C

You are asked to roll a die. Event \begin{align*}A\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*}

### Vocabulary

The possible results of 1 trial of a probability experiment are called ** outcomes**, and the set of all possible outcomes of an event or group of events is the

**. A**

*sample space***is a diagram of overlapping shapes that shows the relationships among members of different sets, and such a diagram helps us to find**

*Venn diagram***, or the chance that something will happen.**

*probability*### Guided Practice

Let’s say our sample space is the numbers from 1 to 10. Event \begin{align*}A\end{align*}

**Answer:**

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*}

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

### Interactive Practice

### Practice

- \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*}?