HSS.CP.A.1Common CoreMathStatistics and ProbabilityGrades 9-12
HSS.CP.A.1: Describing Events with Sample Spaces, Unions, Intersections and Complements
In plain English: HSS.CP.A.1 is the Common Core statistics and probability standard that asks students to describe events as subsets of a sample space, the set of all outcomes, by using characteristics or categories of the outcomes. Students also build new events from old ones with union (or), intersection (and) and complement (not). It is usually taught in Geometry or Algebra II.
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
Common Core State Standards for Mathematics · Domain: Conditional Probability and the Rules of Probability (CP) · Cluster: Understand independence and conditional probability and use them to interpret data Also written as HSS-CP.A.1 or S-CP.1 · Official standard
Students learn the language that the rest of the probability unit depends on. A sample space is the set of all possible outcomes of a chance process, and an event is any subset of it, usually described by a characteristic such as "even", "a senior" or "the sum is 8". Students list the outcomes that fit a description and write the event as a set.
Students then build new events from old ones. The word "or" becomes a union, "and" becomes an intersection, and "not" becomes a complement. Venn diagrams, outcome grids and small data rosters let students see each operation as a region, and they check their lists by making sure every outcome of the sample space is accounted for exactly once.
Learning Objectives
By the end of this lesson, students will be able to:
Write the sample space of a chance process as a set of outcomes
Describe an event as a subset of the sample space using a characteristic or category of the outcomes
Translate "or", "and" and "not" into the union A ∪ B, the intersection A ∩ B and the complement Aᶜ, and list the outcomes of each
Describe a given set of outcomes in words and in set notation
Use Venn diagrams and outcome grids to show events and combinations of events
Prior Knowledge Required
Students should already be comfortable with:
Listing the outcomes of compound events with organized lists, tables and tree diagrams 7.SP.C.8
Uniform probability models and the probability of a simple event 7.SP.C.7
Give students two minutes to answer on their own, then compare with a partner.
Warm-Up Prompt
"A number cube is rolled once. Write every outcome that fits each description: (1) the number is prime, (2) the number is greater than 4, (3) the number is prime or greater than 4, (4) the number is prime and greater than 4."
Answers: (1) {2, 3, 5}, (2) {5, 6}, (3) {2, 3, 5, 6}, (4) {5}. The discussion point is item (3). Some students leave out 5 because it fits both descriptions, which treats "or" as "one or the other but not both". Ask whether a roll of 5 makes the statement "prime or greater than 4" true. It does, so in probability "or" always includes outcomes that fit both. Keep the four lists on the board: they are the first examples of an event, a union and an intersection.
Direct Instruction20 minutes
Vocabulary. An outcome is one possible result. The sample space S is the set of all outcomes. An event is a subset of S, and it "happens" when the outcome is one of its members. Events are usually named by a characteristic of the outcomes. Then introduce the three ways to make new events:
Union, A ∪ B, read "A or B": every outcome in A, in B, or in both.
Intersection, A ∩ B, read "A and B": only the outcomes in both A and B.
Complement, Aᶜ (also written A′), read "not A": every outcome of S that is not in A. A and Aᶜ together make up S, with no outcome in both.
Check the count: every outcome of S belongs to exactly one region of the Venn diagram, so the region counts add up to the size of S.
Work through the examples below. Examples 2-4 use Diagram 1, where S is a set of 20 raffle tickets, A = "the ticket number is even" and B = "the ticket number is a multiple of 3". For each example, ask students to say the event in words before they write the set.
Event from a characteristic
A coin is flipped three times, so S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Describe the event "at least two heads" as a subset of S.
Equation: {HHH, HHT, HTH, THH}, 4 of the 8 outcomes
Intersection ("and")
Tickets 1-20. List the event "the number is even and a multiple of 3."
Equation: A ∩ B = {6, 12, 18}
Union ("or")
Tickets 1-20. List the event "the number is even or a multiple of 3."
Equation: A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20}, 13 outcomes
Complement ("not")
Tickets 1-20. List the event "the number is not a multiple of 3."
Two number cubes are rolled (Diagram 2 shows all 36 outcomes). Describe the event "not (the sum is 10 or less)" by listing its outcomes.
Equation: The sum is 11 or 12: {(5, 6), (6, 5), (6, 6)}, 3 outcomes
After Example 3, point out that 6, 12 and 18 are listed once even though they are in both events: a set never lists a member twice. After Example 5, show that it is faster to find the complement first (the sum is more than 10) than to list the 33 outcomes whose sum is 10 or less. Use Diagram 2 to show that an event described in words is simply a set of cells in the grid.
Guided Practice15 minutes
Display this invented roster of 10 students. The sample space is the set of 10 students, and each student has two characteristics.
Invented roster of 10 students
Student
Grade
Plays a sport
Ana
10
Yes
Ben
11
No
Chloe
11
Yes
Dev
10
Yes
Elena
11
No
Farid
10
No
Grace
11
Yes
Hugo
10
No
Iris
11
No
Jalen
10
Yes
Let G = "the student is in grade 11" and P = "the student plays a sport". Pairs list each event, then place the 10 names in a two-circle Venn diagram:
G = {Ben, Chloe, Elena, Grace, Iris} and P = {Ana, Chloe, Dev, Grace, Jalen}
G ∩ P = {Chloe, Grace}
G ∪ P = {Ana, Ben, Chloe, Dev, Elena, Grace, Iris, Jalen}
Pᶜ = {Ben, Elena, Farid, Hugo, Iris}
(G ∪ P)ᶜ = {Farid, Hugo}: in grade 10 and no sport
Then reverse the task: say "{Ana, Dev, Jalen}" and ask pairs to describe it in words and symbols (grade 10 and plays a sport, Gᶜ ∩ P). Watch for pairs that list Chloe and Grace twice in G ∪ P, and for pairs that find Pᶜ by looking only at grade 11.
Independent Practice10-15 minutes
A spinner has 8 equal sections numbered 1-8. Let A = "odd", B = "greater than 5" and C = "a perfect square". Students list A, B and C, then A ∩ B, A ∪ B, Aᶜ, B ∩ C and A ∩ Cᶜ, and describe each one in words. Answers: A = {1, 3, 5, 7}, B = {6, 7, 8}, C = {1, 4}, A ∩ B = {7}, A ∪ B = {1, 3, 5, 6, 7, 8}, Aᶜ = {2, 4, 6, 8}, B ∩ C = ∅ (no outcome is both greater than 5 and a perfect square), and A ∩ Cᶜ = {3, 5, 7}. Use B ∩ C to introduce the empty set: an event can have no outcomes.
Closure5-10 minutes
Exit ticket: S = {1, 2, 3, ..., 10}. M = "a multiple of 4" and L = "less than 5". (1) List M ∪ L, M ∩ L and Mᶜ. (2) Describe Lᶜ in words. Answers: M ∪ L = {1, 2, 3, 4, 8}, M ∩ L = {4}, Mᶜ = {1, 2, 3, 5, 6, 7, 9, 10}, and Lᶜ is "the number is 5 or more".
Differentiation Strategies
For Struggling Students
Give a blank two-circle Venn diagram with the four regions labeled "A only", "both", "B only" and "neither", and have students place every outcome before answering any question
Post a three-row anchor chart: or = union = shade both circles, and = intersection = shade only the overlap, not = complement = shade everything outside
Start with sample spaces of 6-10 outcomes that students can list in full before moving to grids of 36
For Advanced Students
Ask students to test with several examples whether (A ∪ B)ᶜ always equals Aᶜ ∩ Bᶜ, and whether (A ∩ B)ᶜ always equals Aᶜ ∪ Bᶜ, and to explain each result with a Venn diagram
Give three events A, B and C and ask for the eight regions of a three-circle Venn diagram, each described in symbols
Ask students to write a sample space and two events for which A ∩ B = A, and to explain what that says about A and B
Assessment Guidance
What to Look For
Check that students name the sample space before listing any event, and that every event they write is a subset of it. When students form a union, look for outcomes in both events listed once, not twice or not at all. For complements, check that students take everything in S outside the event, not the "opposite" category they think of first. Ask students to read a symbol such as A ∩ Bᶜ aloud in words ("A and not B"); being able to move between words, symbols and a shaded region is the core of this standard.
02
Classroom Activities
3 Activities
1
Venn Sort with Number Cards
20 minPairs
Pairs sort 24 index cards numbered 1-24 into a two-circle Venn diagram drawn on paper, then read unions, intersections and complements straight from the regions.
Procedure
Each pair draws a large rectangle labeled S with two overlapping circles, and receives cards numbered 1-24
Round 1 (modeled together): A = "multiple of 4" and B = "greater than 15". Pairs place every card, then list A ∩ B = {16, 20, 24}; A ∪ B has 12 cards (4, 8, 12 and 16-24); and (A ∪ B)ᶜ has the other 12 cards
Rounds 2-4: partners take turns choosing two events from this list: odd, a perfect square, has the digit 1, a multiple of 3, less than 10
For each round, pairs write A ∪ B, A ∩ B, Aᶜ and (A ∪ B)ᶜ as sets and check that the four region counts add to 24
Discussion Questions
Which region of the diagram is A ∪ B? Which cards are in A ∪ B but not in A ∩ B?
No two events on the list have an empty A ∩ B. Name an event you could pair with "odd" so that A ∩ B is empty, and describe what the diagram would look like.
Why do the counts in the four regions always add to 24?
Modification for Distance Learning
Use a shared slide with the numbers 1-24 as movable text boxes and a Venn diagram drawn on the background. Pairs drag the numbers into place, then type the sets in the speaker notes.
2
Two-Cube Grid Coloring
20 minPairs
Each pair builds the 36-outcome grid for two number cubes, colors events described in words, and then works backward from a colored set to a description.
Procedure
Pairs draw a 6-by-6 grid like Diagram 2, with the first cube down the side and the second across the top
Color R = "the first cube is even" in one color and T = "the sum is 7" in a second color. Circle the cells that have both colors: R ∩ T = {(2, 5), (4, 3), (6, 1)}
Count R ∪ T (21 cells) by counting colored cells once each, and Tᶜ (30 cells) by counting uncolored-by-T cells
Each partner then colors a secret event on a fresh grid and trades; the other partner writes the event in words and as a union, intersection or complement if possible
Finish by rolling the cubes 10 times and recording, for each roll, whether R, T, R ∩ T and R ∪ T happened
Discussion Questions
Why is (2, 5) a different outcome from (5, 2)?
How can you count R ∪ T without counting the three circled cells twice?
On a roll where R ∩ T happened, did R ∪ T also happen? Is the reverse always true?
Challenge Variation
Ask pairs to find two events on the grid that have no cells in common but together fill the whole grid. What is the relationship between them? (Each is the complement of the other.)
3
Words-to-Symbols Card Match
15 minGroups of 3-4
Groups match 6 word cards to 6 symbol cards about one card drawn from a standard 52-card deck, with H = "the card is a heart" and F = "the card is a face card (jack, queen or king)", then count each event with a real deck.
The 12 Cards
Word cards: "a heart or a face card"; "a heart that is a face card"; "not a heart"; "a face card that is not a heart"; "neither a heart nor a face card"; "not both a heart and a face card"
Symbol cards: H ∪ F; H ∩ F; Hᶜ; F ∩ Hᶜ; (H ∪ F)ᶜ; (H ∩ F)ᶜ
Key, with counts: H ∪ F has 22 cards, H ∩ F has 3, Hᶜ has 39, F ∩ Hᶜ has 9, (H ∪ F)ᶜ has 30, (H ∩ F)ᶜ has 49
Procedure
Groups match the cards, then sort a real deck into piles to count each event
Each group explains one match to the class using a Venn diagram with the regions "hearts only", "heart face cards", "face cards only" and "neither"
Groups check that 10 + 3 + 9 + 30 = 52
Discussion Questions
Why are "neither a heart nor a face card" and "not both a heart and a face card" different events?
Which two events on the cards are complements of each other?
Modification for Distance Learning
Put the 12 cards in a shared drag-and-drop matching slide and show a picture of the 52 cards laid out in four rows by suit so students can count.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Venn Diagram of Two Events in One Sample Space
Each of the 20 tickets appears exactly once. A = "even" is the left circle and B = "multiple of 3" is the right circle. The overlap is A ∩ B, both circles together are A ∪ B, and everything outside the left circle is Aᶜ. Worked Examples 2-4 read their answers from this picture.
Diagram 2: The Two-Cube Sample Space as a Grid
Rows give the first cube and columns give the second, so each cell is one outcome. An event described in words ("the sum is 8", "doubles") is a set of cells. The intersection is the set of cells in both events, and the complement of D is the 31 unshaded cells.
04
Homework Assignment
~30 min
HSS.CP.A.1 Homework: Describing Events
Directions: For every problem, write the sample space or say how many outcomes it has before you list any event. Write events as sets in braces, and describe each event in words as well as in symbols. Check that no outcome is listed twice.
Part 1: Sample Spaces and Events (Problems 1-2)
Raffle tickets numbered 21 through 35 are in a bowl, and one is drawn. (a) Write the sample space and state how many outcomes it has. (b) List the outcomes in F = "the number is a multiple of 5", G = "the digits of the number add to 7" and H = "the number is odd".
A coin is flipped four times. (a) Write the sample space (it has 16 outcomes, such as HTTH). (b) List the event "exactly three heads". (c) List the event "the first and last flips match" and state how many outcomes it has.
Part 2: Union, Intersection and Complement (Problems 3-4)
Use the events F, G and H from Problem 1. List F ∪ G, F ∩ G, Hᶜ and F ∩ Hᶜ, and describe each one in words.
In a group of 25 students, 14 take Spanish (event S), 8 take art (event A), and 5 take both. (a) Draw a Venn diagram and write the number of students in each of the four regions. (b) How many students are in S ∪ A, in S ∩ Aᶜ and in (S ∪ A)ᶜ? Describe each of these events in words.
Part 3: Words and Symbols (Problems 5-6)
For a student chosen at a school, let E = "the student plays an instrument" and R = "the student rides the bus". (a) Write in symbols: plays an instrument but does not ride the bus; rides the bus or plays an instrument; neither plays an instrument nor rides the bus; does not do both. (b) Write Eᶜ ∪ R in words.
Two number cubes are rolled. Let A = "the sum is even" and B = "the product is greater than 20". (a) List B. (b) How many outcomes are in A, in A ∩ B, in A ∪ B and in Bᶜ? (c) List A ∩ B.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sample Space
Complete and correct, with the number of outcomes
One outcome missing or repeated
Missing or wrong
Events from Characteristics
Every event listed correctly as a subset of S
Most events correct
Events not listed or not subsets of S
Union, Intersection, Complement
All correct, no double listing, complements taken within S
One or two errors
Operations confused
Words and Symbols
Translations correct in both directions
Correct in one direction only
Missing or incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Answer each question before you open its explanation. The score at the top counts your multiple-choice answers, and Reset quiz starts the whole set again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A coin is flipped and a number cube is rolled. How many outcomes are in the sample space?
Answer: C
Each of the 2 coin results pairs with each of the 6 cube results, so there are 2 × 6 = 12 outcomes: H1 through H6 and T1 through T6. Choice A adds 2 + 6 instead of pairing every coin result with every cube result. Choice D is the sample space for two number cubes.
Question 2 of 20 · Multiple Choice
The sample space is the 12 months of the year. Which set is the event "the month has 31 days"?
Answer: B
Seven months have 31 days: January, March, May, July, August, October and December. Choice A assumes the months simply alternate, which breaks after July. Choice C lists the 30-day months, and choice D leaves out December.
Question 3 of 20 · Multiple Choice
Same sample space. Let T = "the month has 31 days" and J = "the name of the month starts with J". Which set is T ∩ J?
Answer: A
The intersection keeps only the months in both events: the J months are January, June and July, and of those only January and July have 31 days. Choice B is J by itself, choice C is the union T ∪ J, and choice D is J ∩ Tᶜ.
Question 4 of 20 · Multiple Choice
Same sample space and event T. How many outcomes are in Tᶜ?
Answer: D
Tᶜ is every month that does not have 31 days: February, April, June, September and November, so 12 - 7 = 5. Choice C counts only the 30-day months and forgets February. Choice A counts T itself, and choice B is the whole sample space.
Question 5 of 20 · Multiple Choice
S = {1, 2, 3, ..., 15}. Let A = "the number is odd" and B = "the number is a multiple of 5". How many outcomes are in A ∪ B?
Answer: B
A = {1, 3, 5, 7, 9, 11, 13, 15} and B = {5, 10, 15}. The union is {1, 3, 5, 7, 9, 10, 11, 13, 15}, which has 9 outcomes: only 10 is added to A. Choice A adds 8 + 3 and counts 5 and 15 twice. Choice C counts A ∩ B, and choice D counts A alone.
Question 6 of 20 · Multiple Choice
For a student chosen at random, E = "the student is a senior". Which notation describes "the student is not a senior"?
Answer: C
The complement Eᶜ contains every outcome of the sample space that is not in E. Choices A and B combine E with the whole sample space S, which gives S and E, not "not E". Choice D, the empty set, is an event with no outcomes.
Question 7 of 20 · Multiple Choice
For a student at a school, D = "the student owns a dog" and K = "the student owns a cat". Which description matches D ∩ K?
Answer: D
An intersection means "and": the outcome must be in both events, so the student owns both pets. Choice A describes the union D ∪ K. Choice B is D ∩ Kᶜ, and choice C is (D ∪ K)ᶜ.
Question 8 of 20 · Multiple Choice
Maya is on the soccer team (event S) and in the school band (event B). Which statement is true?
Answer: A
Maya is in both events, so she is in the intersection, and every outcome of an intersection is also in the union. Choice B treats "or" as "one but not both", which is not how probability uses "or". Choice C is impossible, because S ∩ B is always inside S ∪ B.
Question 9 of 20 · Multiple Choice
A coin is flipped twice, so S = {HH, HT, TH, TT}. Which event is the complement of "at least one tail"?
Answer: C
"At least one tail" is {HT, TH, TT}. The complement is every other outcome of S, which is {HH}: no tails at all. Choice A confuses "not at least one tail" with "all tails". Choices B and D are not the outcomes left over when {HT, TH, TT} is removed from S.
Question 10 of 20 · Multiple Choice
A coin is flipped and a number cube is rolled. Which set is the event "heads and an even number"?
Answer: A
Each outcome must show heads and an even number, so the event is {H2, H4, H6}. Choice B is the union "heads or an even number". Choice C is the event "an even number" and ignores the coin. Choice D is "tails and an odd number", the complement of the union in choice B.
Question 11 of 20 · Multiple Choice
In a homeroom of 30 students, 12 are in chess club (C), 9 are in robotics (R), and 4 are in both. How many students are in (C ∪ R)ᶜ?
Answer: C
The Venn regions are 8 chess only, 4 both and 5 robotics only, so C ∪ R has 17 students and (C ∪ R)ᶜ has 30 - 17 = 13. Choice A is C ∪ R itself. Choice B computes 30 - 12 - 9, which subtracts the 4 students in both clubs twice. Choice D is the number in robotics only.
Question 12 of 20 · Multiple Choice
S = {1, 2, 3, ..., 12}. Which event is the empty set ∅?
Answer: B
Every multiple of 4 is even, so no number is both odd and a multiple of 4, and the intersection has no outcomes. Choice A is {2}, since 2 is even and prime. Choices C and D both describe all 12 outcomes, which is S, not ∅.
Question 13 of 20 · Multiple Choice
Which expression describes the event "neither A nor B"?
Answer: D
"Neither A nor B" means the outcome is not in A or B, which is the complement of the union: (A ∪ B)ᶜ, the region outside both circles. Choice A means "not both", which still includes outcomes in exactly one of the events. Choice C means "A but not B".
Question 14 of 20 · Multiple Choice
A lunch special is one sandwich (turkey, veggie or ham) and one drink (milk or juice), so the sample space has 6 lunches. How many lunches are in the event "veggie or juice"?
Answer: A
Veggie lunches: veggie-milk and veggie-juice. Juice lunches: turkey-juice, veggie-juice and ham-juice. Veggie-juice is in both, so the union has 2 + 3 - 1 = 4 lunches. Choice B counts veggie-juice twice. Choice C is the intersection "veggie and juice".
Question 15 of 20 · Short Answer
S = {1, 2, 3, ..., 16}. Let A = "the number is a perfect square" and B = "the number is even". List A ∩ B, A ∪ B and Aᶜ.
A = {1, 4, 9, 16} and B = {2, 4, 6, 8, 10, 12, 14, 16}. A ∩ B = {4, 16}. A ∪ B = {1, 2, 4, 6, 8, 9, 10, 12, 14, 16}, 10 outcomes. Aᶜ = {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15}, the 12 numbers that are not perfect squares.
Question 16 of 20 · Short Answer
The sample space is the 7 days of the week. Let W = "the day is on the weekend" and T = "the name of the day starts with T". Describe each set as an event, in words and with symbols: (a) {Monday, Tuesday, Wednesday, Thursday, Friday} (b) {Tuesday, Thursday, Saturday, Sunday}
(a) The day is not on the weekend: Wᶜ. (b) The day starts with T or is on the weekend: T ∪ W. Other correct descriptions are fine if they give exactly the same set.
Question 17 of 20 · Short Answer
Two number cubes are rolled. How many of the 36 outcomes are in the event "at least one cube shows a 1"? Use the complement to find the answer, and explain.
The complement is "neither cube shows a 1": each cube shows 2-6, so there are 5 × 5 = 25 such outcomes. The event itself has 36 - 25 = 11 outcomes: (1, 1) through (1, 6) and (2, 1) through (6, 1).
Question 18 of 20 · Short Answer
Explain why A ∩ B is always a subset of A ∪ B. Then describe what must be true of A and B for A ∩ B and A ∪ B to be the same event, and give an example.
Every outcome in A ∩ B is in A, so it is also in A ∪ B. The two events are the same only when every outcome that is in A or B is in both, which means A and B are the same event. Example: on a number cube, A = "even" and B = {2, 4, 6} give A ∩ B = A ∪ B = {2, 4, 6}. If A has an outcome that B does not, that outcome is in A ∪ B but not in A ∩ B.
Question 19 of 20 · Short Answer
An animal shelter lists 8 animals (invented data): Max (dog, under 1 year), Luna (cat, 1 year or older), Bo (dog, 1 year or older), Pip (cat, under 1 year), Rex (dog, 1 year or older), Mia (cat, 1 year or older), Taz (dog, under 1 year) and Zoe (cat, under 1 year). One animal is chosen. Let D = "the animal is a dog" and Y = "the animal is under 1 year old". List D ∩ Yᶜ, D ∪ Y and (D ∪ Y)ᶜ, and describe each in words.
D ∩ Yᶜ = {Bo, Rex}: dogs 1 year or older. D ∪ Y = {Max, Bo, Pip, Rex, Taz, Zoe}: a dog or a young animal (6 animals). (D ∪ Y)ᶜ = {Luna, Mia}: neither a dog nor under 1 year, so cats 1 year or older.
Question 20 of 20 · Short Answer
A spinner with three equal sections (red, blue, green) is spun twice. (a) List the sample space. (b) List the event "the same color both times" and state how many outcomes are in its complement. (c) How many outcomes are in "red on the first spin or red on the second spin"?
(a) S = {RR, RB, RG, BR, BB, BG, GR, GB, GG}, 9 outcomes. (b) {RR, BB, GG}; its complement has 6 outcomes. (c) {RR, RB, RG, BR, GR}: 5 outcomes, with RR counted once.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSS.CP.A.1 mean?
HSS.CP.A.1 means students can describe events as sets of outcomes. They list the sample space, pick out the outcomes that share a characteristic (such as "even" or "a senior"), and combine events with "or" (union), "and" (intersection) and "not" (complement). It is the vocabulary standard for the whole conditional probability domain.
Is HSS.CP.A.1 taught in Geometry or Algebra 2?
It depends on the course sequence. The traditional pathway in the Common Core appendix on course design places the conditional probability standards in Geometry, and the integrated pathway places them in Mathematics II. Some schools teach them in Algebra II or in a statistics course instead.
What is the difference between an outcome, a sample space and an event?
An outcome is one result, the sample space is the set of all results, and an event is a subset of the sample space. When a number cube is rolled, 4 is an outcome, {1, 2, 3, 4, 5, 6} is the sample space, and "the number is even", {2, 4, 6}, is an event. An event can have one outcome, many, all of them or none.
Does "or" in probability include outcomes that are in both events?
Yes. In probability, "A or B" is the union and includes outcomes in A, in B, or in both. Everyday speech sometimes uses "or" to mean one but not both, which is a frequent source of errors. If a problem means "exactly one of the two", it has to say so.
What is the complement of an event?
The complement of A is every outcome of the sample space that is not in A. It is written Aᶜ, A′ or Ā. Complements depend on the sample space: for a number cube the complement of {6} is {1, 2, 3, 4, 5}, but for a 12-sided die it has 11 outcomes. Finding a complement first is often the quickest way to list or count an event such as "at least one".
What symbols are used for union, intersection and complement?
Union: A ∪ B, read "A or B"
Intersection: A ∩ B, read "A and B"
Complement: Aᶜ, A′ or Ā, read "not A"
Empty set: ∅ or { }, an event with no outcomes
A memory aid some teachers use: ∪ looks like a cup that holds everything from both sets.
What mistakes do students make with unions and intersections?
Three errors come up often. Students list or count the shared outcomes twice in a union; they read "or" as "one but not both"; and they take a complement as the "opposite category" they have in mind instead of everything else in the sample space. A Venn diagram with every outcome placed exactly once prevents all three.
How does HSS.CP.A.1 connect to later probability standards?
The later standards in the domain are written in its language. HSS.CP.A.2 defines independence with P(A and B), HSS.CP.A.3 defines conditional probability with P(A and B)/P(B), and HSS.CP.B.7 is the Addition Rule for P(A or B). Students who cannot yet read A ∩ B as an event struggle with all three.
Can an event be empty, or be the whole sample space?
Yes to both. An event with no outcomes, such as "a number cube shows 7", is the empty set and has probability 0. The whole sample space is also an event, and it always happens. Two events with an empty intersection, such as "less than 3" and "greater than 4" on a number cube, cannot happen on the same trial.
How can students practice describing events at home?
Use a deck of cards, a pair of dice or a family calendar. One person names two characteristics, for example "a red card" and "a card less than 5", and the other says or writes the union, intersection and complement, then checks by sorting the cards. Asking "what is the sample space?" before every answer builds the habit this standard needs.
07
Related Standards
5 standards
These standards connect to HSS.CP.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.SP.C.8Prerequisite
Find probabilities of compound events using organized lists, tables, tree diagrams and simulation
