# MA.912.DP.4.1

Describe events as subsets of a sample space using characteristics, or categories, of the outcomes, or as unions, intersections or complements of other events.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Data Analysis and Probability
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Event
• Sample space

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In middle grades, students began to identify sample spaces and explored basic probability by using theoretical probability for repeated experiments. In Mathematics for College Liberal Arts, students break the sample space up into subsets. Students also identify the union, intersection, and complement of other events.
• Within this benchmark, students should use their knowledge of Venn Diagrams and set notation from the Logic and Discrete Theory (LT) benchmarks within their work of these Data Analysis and Probability (DP) benchmarks. The expectation is students are working with multiple events (2 or more).
• Instruction includes finding the sample space for an event and then subsets of that sample space.
• For example, the sample space for rolling a die is S = {1, 2, 3, 4, 5, 6}. The sample space for rolling an even number, which is a subset, is S = {2, 4, 6}.
• Instruction includes the use of diagrams in order to explore and illustrate concepts of complements, unions, intersections, differences and products of two sets.
• The intersection of two sets is denoted as A B and consists of all elements that are both in A and B
• The complement of a set can be denoted as Ac, -A-, A′ or ~A. This is the set of all elements that are in the universal set S but not in A. Complements can be of unions or intersections.
• Instruction includes finding sample spaces for multiple events.
• For example, find the sample space of rolling an even number on a six-sided die and flipping a head on a coin.
• Sample space for rolling a six-sided die: R = {1, 2, 3, 4, 5, 6}.
• Sample space for rolling an even number on a six-sided die: E = {2, 4, 6}.
• Sample space for flipping a coin: F = {H, T}.
• Sample space for flipping a head: H = {H}.
• Sample space for rolling a die and flipping a coin: S = {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}.
• Sample space for rolling an even number and flipping a head: A = {2H, 4H, 6H}.
• Sample space for not rolling an even number and not flipping a head: B = {1T, 3T, 5T}.
• Instruction includes finding the sample space of union and intersections.
• For example, a deck of 16 cards has 4 red hearts, 4 red diamonds, 4 black spades, and 4 black clubs, numbered 1 to 4.
• Sample Space for the deck of cards: S = {1H, 2H, 3H, 4H, 1D, 2D, 3D, 4D, 1S, 2S, 3S, 4S, 1C, 2C, 3C, 4C}.
• Sample Space for the intersection of red and 3s:
• Define the subset of Red: R = {1H, 2H, 3H, 4H, 1D, 2D, 3D, 4D}.
• Define the subset of 3s: 3s = {3H, 3D, 3S, 3C}.
• Define the intersection of Red and 3s: R ∩ 3S = {3H, 3D}.
• Sample Space for the union of clubs and not diamonds:
• Define the subset of clubs: C = {1C, 2C, 3C, 4C}.
• Define the subset of not diamonds: D′ = {1H, 2H, 3H, 4H, 1D, 2D, 3D, 4D, 1S, 2S, 3S, 4S, 1C, 2C, 3C, 4C}.
• Define the union of clubs and not diamonds: C ∪ D′ = {1H, 2H, 3H, 4H, 1D, 2D, 3D, 4D, 1S, 2S, 3S, 4S, 1C, 2C, 3C, 4C}.

### Common Misconceptions or Errors

• If finding the unions when the events are not mutually exclusive, students often count the intersection more than once because it is included in multiple sets.
• Students may forget about the values that are not in the sets.
• Students may confuse the symbols used in basic set notation

• The table below provides information on 10 students within Student Government. For each student, their sex, age, whether or not they plan to go to college, if they play a sport, and how many honors courses they are currently enrolled in.

• Part A. What is the sample space for number of colleges these students plan to apply to?
• Part B . What outcomes from the sample space in part A are in the event that the student plans to study business?
• Part C . Consider the following 3 events . For each list , which students would be make up the event:
• F = The selected student is female
• S = The selected student plays a sport
• A = The selected student is less than 18 years old
• Part D . Based on part C, which outcomes are in the following events?
• FS
• (F S)′
• FA
• Ac

### Instructional Items

Instructional Item 1
• What is the sample space of not rolling a number greater than 4 on a six-sided die?

Instructional Item 2
• What is the sample space of not rolling a number greater than 4 on a six-sided die and flipping a head on a coin?

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1210300: Probability and Statistics Honors (Specifically in versions: 2014 - 2015, 2015 - 2019, 2019 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1210305: Mathematics for College Statistics (Specifically in versions: 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))
1212300: Discrete Mathematics Honors (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.DP.4.AP.1: Given a sample space, select a subset of the sample space or given two sets, select the union, intersection, or complement of two sets.
MA.912.DP.4.AP.2: Given the probability of events A and B and the product of their probabilities, select whether the events are independent or not independent.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Lesson Plans

What Are the Chances?:

Students will develop a program to simulate repeated rolls of a pair of dice in this lesson plan. They will program a realistic interaction between the user and the simulation as well as an analysis tool to identify the theoretical probability and track the observed probability for each outcome.

Type: Lesson Plan

Taxes using Venn Diagrams, Lesson 1:

Students will review constructing and solving Venn diagrams with two and three data sets. Students will then convert text about the collection of taxes from the local, state, and federal governments into a Venn diagram. This is lesson 1 of a three-part integrated mathematics and civics mini-unit.

Type: Lesson Plan

Taxes using Venn Diagrams, Lesson 2:

Students will discuss, recognize, and be challenged to list unions, intersections, and complements related to a Venn diagram created by three data sets. The data is the type of taxes assessed to citizens by the local, state, and federal governments. This is the second lesson in a 3-part integrated mathematics and civic mini-unit.

Type: Lesson Plan

Casino Royale:

Students examine games of chance to determine the difference between dependent and independent conditional probability.

Type: Lesson Plan

The task is intended to address sample space, independence, probability distributions and permutations/combinations.

The Titanic 1:

This task asks students to calculate probabilities using information presented in a two-way frequency table.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

The task is intended to address sample space, independence, probability distributions and permutations/combinations.

The Titanic 1:

This task asks students to calculate probabilities using information presented in a two-way frequency table.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.