Events — Class 12 Statistics Notes (GSEB)

What is an Event?

Event

A subset of the sample space of a random experiment is called an event. Events are generally denoted by letters A, B, C, … or as A₁, A₂, A₃, …. The set formed by the sample points showing favourable outcomes of an event A will be a subset of the sample space U. Thus, any event A associated with the random experiment is the subset of sample space U. This is denoted as A ⊂ U.

Event as a Subset — see an event highlight inside U

U = {(i, j); i, j = 1, 2, 3, 4, 5, 6} for two balanced dice. Click a preset to highlight exactly the cells of that event.

1,1

1,2

1,3

1,4

1,5

1,6

2,1

2,2

2,3

2,4

2,5

2,6

3,1

3,2

3,3

3,4

3,5

3,6

4,1

4,2

4,3

4,4

4,5

4,6

5,1

5,2

5,3

5,4

5,5

5,6

6,1

6,2

6,3

6,4

6,5

6,6

A = "obtaining a perfect square as the number on the upper side" of a die → A = {1, 4} (A shaded inside U).

Events in the random experiment of throwing two balanced dice:

A₁ = the sum of the numbers on the dice is 6 → A₁ = {(1,5), (2,4), (3,3), (4,2), (5,1)}
A₂ = the numbers on the dice are the same → A₂ = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
A₃ = the sum of the numbers on the dice is more than 9 → A₃ = {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)}

All these subsets are called events.

The Event Family — Ten Special Events

Ten special events every examiner loves. Click a card to expand its definition, formula and example — only one stays open at a time.

English → Set Operation — the keyword map

NOT → complement A′AND / both → intersection A ∩ BOR / at least one → union A ∪ BONLY one → difference A − B, B − A

Set Identities and the Venn Lab

Difference — only A happens ⭐

A - B = A \cap B' = A - (A \cap B) = (A \cup B) - B

Difference — only B happens ⭐

B - A = A' \cap B = B - (A \cap B) = (A \cup B) - A

Two dice (36 outcomes)

A = first die shows an even number · B = second die shows an even number

n(U) = 36

1,1

1,2

1,3

1,4

1,5

1,6

2,1

2,2

2,3

2,4

2,5

2,6

3,1

3,2

3,3

3,4

3,5

3,6

4,1

4,2

4,3

4,4

4,5

4,6

5,1

5,2

5,3

5,4

5,5

5,6

6,1

6,2

6,3

6,4

6,5

6,6

A only B only A ∩ B outside

Set notation

Click a region button to shade it and read its count.

Solved Illustrations 6–10

Solved Example

Problem

Illustration 6 (⭐ board pattern): 3 yellow (Y₁, Y₂, Y₃) and 2 pink (P₁, P₂) flowers are in a basket; one flower is randomly selected. A = yellow, B = pink. Find U, A, B, A′, B′, A ∩ B, A ∪ B, A ∩ B′, A′ ∩ B, the elementary events, and state whether A and B are mutually exclusive and exhaustive with reasons.

Solution

A and B are mutually exclusive because A ∩ B = φ, and exhaustive because A ∪ B = U.

Solved Example

Problem

Illustration 7: A = {1, 2, 3, 4}, B = {−1, 0, 1}, U = A ∪ B = {−1, 0, 1, 2, 3, 4}. Find B′, A′ ∩ B and A − B.

Solution

B′ = {2, 3, 4}; A′ ∩ B = {−1, 0}; A − B = {2, 3, 4}.

Solved Example

Problem

Illustration 8 (⭐ board pattern): one number is randomly selected from the first 50 natural numbers. U = {1, 2, 3, …, 50}. A = multiple of 5, B = multiple of 7. Find the probability-related sets: multiple of 5 or 7; multiple of both 5 and 7; multiple of 5 but not 7; only a multiple of 7 out of 5 and 7.

Solution

A ∪ B has 16 numbers; A ∩ B = {35}; A ∩ B′ has 9 numbers; A′ ∩ B has 6 numbers.

Solved Example

Problem

Illustration 9: A₁ = {x | x = −1, 0, 1} = {−1, 0, 1}, A₂ = {1, 2, 3}. Find A₁ ∪ A₂ and A₁ ∩ A₂.

Solution

A₁ ∪ A₂ = {−1, 0, 1, 2, 3}; A₁ ∩ A₂ = {1}.

Solved Example

Problem

Illustration 10 (⭐ board pattern — screws, interval form): the length (cm) of a screw is x; A₁ = {x | 0 < x < 1}, A₂ = {x | ½ ≤ x < 2}. Find A₁ ∪ A₂ and A₁ ∩ A₂.

Solution

A₁ ∪ A₂ = (0, 2); A₁ ∩ A₂ = [½, 1).

Practice Checkpoint

Your turn. Work each question in your notebook — type only the final answer here.

Type 2 Q1

3 marks

A sample space of a random experiment of selecting a number is U = {1, 2, 3, 4, …, 20}. Show the numbers representing the following events: (1) the number is an even number (2) the number is divisible by 3 (3) the number is divisible by 2 or 3.

Try the experiment:

From 1 to 20 (20 numbers)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Type 2 Q8

2 marks

Three female employees and two male employees work in an office. An employee from the office staff is randomly selected for training. If the event that the employee selected for training is female is denoted by A and the event that the employee is male is denoted by B, find the sets U, A, B, A ∪ B, A ∩ B and answer: (1) are A and B mutually exclusive? Give a reason (2) are A and B exhaustive? Give a reason.

Try the experiment:

Female ×3Male ×2

Draw from the bag — every item is equally likely.

Type 2 Q9

3 marks

A card is randomly selected from a pile of 52 cards. If event A is that a black card is drawn and event B is that a card from one to ten (not face) is drawn, find the events U, A, B, A ∩ B, A ∪ B and B′.

Try the experiment:

♠

♣

♥

♦

The drawn card light up in the 52-card deck. Draws are with replacement (the deck is unchanged).

Finish the rest of this question type → Practice Page· then return here for the next topic

Key Takeaways

An event is a subset of the sample space: A ⊂ U; the impossible event is φ and the certain event is U.
The complementary event is A′ = U − A (non-occurrence of A).
A ∩ B = both occur; A ∪ B = at least one occurs; A − B = A ∩ B′ = only A occurs.
Mutually exclusive events have A ∩ B = φ; exhaustive events have A ∪ B = U; elementary events (single sample points) are mutually exclusive and exhaustive.
Keyword map: NOT → complement, AND → intersection, OR / at least one → union, ONLY → difference.
The card-pack example: heart or king = 16 cards (A ∪ B with A = 13 hearts, B = 4 kings, A ∩ B = 1 card — the heart king).