Complementary Events Sample Space Quiz (HSS.CP.A.1)

1) If event P represents students who chose Pop as their favorite, how many students are in P′?

35

65

100

45

100 total − 35 pop = 65 not pop.

Explanation

100 total − 35 pop = 65 not pop.

2) If a sample space has n total outcomes and event K has k outcomes, which expression represents the number of outcomes in K′?

N + k

N − k

N × k

K − n

Complement outcomes = total outcomes − event outcomes → n − k.

Explanation

Complement outcomes = total outcomes − event outcomes → n − k.

3) When rolling two dice, let event D = {sum is greater than 9}. Which best describes D′?

Sum is 9 or less

Sum is less than 9

Sum is exactly 9

Sum is greater than or equal to 9

Complement of “sum > 9” is “sum ≤ 9,” or 9 or less.

Explanation

Complement of “sum > 9” is “sum ≤ 9,” or 9 or less.

4) A sample space for selecting a letter from the word “MATHEMATICS” contains the outcomes S = {M, A, T, H, E, I, C, S}. If event V = {vowels}, what is V′?

{A, E, I}

{M, A, T, H, E, M, A, T, I, C, S}

{M, T, H, C, S}

{A, E, I, O, U}

Vowels are {A,E,I}, so complement = {M,T,H,C,S}.

Explanation

Vowels are {A,E,I}, so complement = {M,T,H,C,S}.

5) Which Venn diagram characteristic is true for complementary events E and E′?

They overlap in the middle

E′ is drawn outside the sample space rectangle

They partition the sample space into two non-overlapping regions

They are drawn as two separate circles that don't touch

E and E′ split the sample space into two parts with no overlap.

Explanation

E and E′ split the sample space into two parts with no overlap.

6) In a sample space with 50 equally likely outcomes, event A contains 18 outcomes. How many outcomes does A′ contain?

18

32

50

68

50 total − 18 in A = 32 in A′.

Explanation

50 total − 18 in A = 32 in A′.

7) If event T = {students who did NOT choose Other}, how many students are in T?

8

71

84

92

100 total − 8 who chose Other = 92 not Other.

Explanation

100 total − 8 who chose Other = 92 not Other.

8) Which statement correctly describes the relationship between event C (students who chose Country) and C′?

C ∪ C′ = {all 100 students surveyed}

C and C′ have 12 outcomes in common

C and C′ are the same set

C ∪ C′ contains only 88 students

An event and its complement together include every student.

Explanation

An event and its complement together include every student.

9) Using the event N from question 12, how many students are in N′?

45

35

100

55

100 − 45 = 55 not rock or hip-hop.

Explanation

100 − 45 = 55 not rock or hip-hop.

10) Let event N = {students who chose Rock or Hip-Hop}. How many outcomes are in N?

25

20

45

55

Rock + Hip-Hop = 25 + 20 = 45.

Explanation

Rock + Hip-Hop = 25 + 20 = 45.

11) A sample space S contains all possible outcomes when rolling a standard six-sided die. If event A = {rolling an even number}, what is the complement of A?

{1, 3, 5}

{2, 4, 6}

{1, 2, 3, 4, 5, 6}

{0, 2, 4, 6}

Even numbers are {2,4,6}, so the complement is {1,3,5}.

Explanation

Even numbers are {2,4,6}, so the complement is {1,3,5}.

12) A spinner has 8 equal sections numbered 1–8. Event M = {multiples of 3}. How many outcomes are in M′?

6

2

3

8

Multiples of 3 are {3,6}, so 8 − 2 = 6 outcomes are not multiples of 3.

Explanation

Multiples of 3 are {3,6}, so 8 − 2 = 6 outcomes are not multiples of 3.

13) If event E and event F are complementary, what is E ∩ F (the intersection of E and F)?

S (the entire sample space)

E

F

∅ (the empty set)

Complementary events have no overlap, so their intersection is empty.

Explanation

Complementary events have no overlap, so their intersection is empty.

14) Which of the following pairs represents complementary events when drawing one card from a standard deck?

Drawing a heart; drawing a diamond

Drawing a face card; drawing a number card

Drawing a king; drawing a queen

Drawing a red card; drawing a black card

Every card is either red or black, not both, so they are complements.

Explanation

Every card is either red or black, not both, so they are complements.

15) When flipping a coin twice, the sample space is S = {HH, HT, TH, TT}. If event C = {getting at least one head}, what is C′?

{HH, HT, TH}

{HH}

{TT}

{HT, TH}

“At least one head” includes HH, HT, TH, so only TT is left as the complement.

Explanation

“At least one head” includes HH, HT, TH, so only TT is left as the complement.

16) A sample space consists of outcomes {1, 2, 3, 4, 5, 6, 7, 8}. Event B = {2, 4, 6, 8}. What is B′?

{2, 4, 6, 8}

{1, 3, 5, 7}

{1, 2, 3, 4, 5, 6, 7, 8}

{}

Complement of {2,4,6,8} includes all other numbers → {1,3,5,7}.

Explanation

Complement of {2,4,6,8} includes all other numbers → {1,3,5,7}.

17) If P(A) represents the probability of event A, and A′ represents the complement of A, which equation must always be true?

P(A) + P(A′) = 1

P(A) = P(A′)

P(A) × P(A′) = 1

P(A) − P(A′) = 0

An event and its complement always add up to 1.

Explanation

An event and its complement always add up to 1.

18) In a Venn diagram showing a sample space S with event E inside, what does the region outside of E but inside S represent?

The intersection of E and S

The union of E and S

The complement of S

The complement of E

The space outside E but inside S shows all outcomes not in E.

Explanation

The space outside E but inside S shows all outcomes not in E.

19) A bag contains red, blue, green, and yellow marbles. If event R represents selecting a red marble, which of the following represents the complement of R?

Selecting a red marble

Selecting a blue marble only

Selecting a blue, green, or yellow marble

Selecting no marble

The complement of “red” means all marbles that are not red → blue, green, and yellow.

Explanation

The complement of “red” means all marbles that are not red → blue, green, and yellow.

20) Which statement best describes complementary events?

Two events that can occur at the same time

Two events that together contain all outcomes in the sample space with no overlap

Two events that have exactly one outcome in common

Two events that are equally likely to occur

Complementary events cover all possible outcomes and never overlap.

Explanation

Complementary events cover all possible outcomes and never overlap.

Representing Complementary Events with Sample Spaces Quiz

1) If event P represents students who chose Pop as their favorite, how many students are in P′?

2)

What first name or nickname would you like us to use?

2) If a sample space has n total outcomes and event K has k outcomes, which expression represents the number of outcomes in K′?

3) When rolling two dice, let event D = {sum is greater than 9}. Which best describes D′?

4) A sample space for selecting a letter from the word “MATHEMATICS” contains the outcomes S = {M, A, T, H, E, I, C, S}. If event V = {vowels}, what is V′?

5) Which Venn diagram characteristic is true for complementary events E and E′?

6) In a sample space with 50 equally likely outcomes, event A contains 18 outcomes. How many outcomes does A′ contain?

7) If event T = {students who did NOT choose Other}, how many students are in T?

8) Which statement correctly describes the relationship between event C (students who chose Country) and C′?

9) Using the event N from question 12, how many students are in N′?

10) Let event N = {students who chose Rock or Hip-Hop}. How many outcomes are in N?

11) A sample space S contains all possible outcomes when rolling a standard six-sided die. If event A = {rolling an even number}, what is the complement of A?

12) A spinner has 8 equal sections numbered 1–8. Event M = {multiples of 3}. How many outcomes are in M′?

13) If event E and event F are complementary, what is E ∩ F (the intersection of E and F)?

14) Which of the following pairs represents complementary events when drawing one card from a standard deck?

15) When flipping a coin twice, the sample space is S = {HH, HT, TH, TT}. If event C = {getting at least one head}, what is C′?

16) A sample space consists of outcomes {1, 2, 3, 4, 5, 6, 7, 8}. Event B = {2, 4, 6, 8}. What is B′?

17) If P(A) represents the probability of event A, and A′ represents the complement of A, which equation must always be true?

18) In a Venn diagram showing a sample space S with event E inside, what does the region outside of E but inside S represent?

19) A bag contains red, blue, green, and yellow marbles. If event R represents selecting a red marble, which of the following represents the complement of R?

20) Which statement best describes complementary events?