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.)

**Grade:**912

**Strand:**Data Analysis and Probability

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### 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}.

- For example, the sample space for rolling a die is
- 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 A
^{c},, A′ or ~A. This is the set of all elements that are in the universal set^{-}A^{-}*S*but not in*A*. Complements can be of unions or intersections.

- The intersection of two sets is denoted as
- 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*}.

- Sample space for rolling a six-sided die:

- For example, find the sample space of rolling an even number on a six-sided die
and flipping a head on a coin.
- 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}*.

- Sample Space for the deck of cards:

- 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.

### 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

### Instructional Tasks

*Instructional Task 1 (MTR.7.1)*

- 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?
*F*∪*S*- (
*F*∪*S*)′ *F*∩*A**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

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

## Problem-Solving Tasks

## Student Resources

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

## Problem-Solving Tasks

Return to Fred's Fun Factory (with 50 cents):

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

Type: Problem-Solving Task

The Titanic 1:

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

Type: Problem-Solving Task

## Parent Resources

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

## Problem-Solving Tasks

Return to Fred's Fun Factory (with 50 cents):

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

Type: Problem-Solving Task

The Titanic 1:

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

Type: Problem-Solving Task