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

1) 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}.

2) 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.

3) 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.

4) 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.

5) 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.

6) 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}.

7) 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.

8) 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.

9) 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.

10) 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.

11) 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.

12) 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.

13) 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.

14) 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.

15) 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.

16) 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′.

17) 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.

18) 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}.