Random Events and the Sample Space

Random Experiments and the Sample Space

Definition of a Random Experiment

Characteristics of a Random Experiment

Characteristics:

1. Repeatability: the experiment can be repeated under the same conditions
2. Uncertainty: the outcome of each trial cannot be determined in advance
3. Observability: the outcome of the experiment can be observed

Definition of the Sample Space

Definition of a Sample Point

Examples

Example 1: the random experiment of tossing a coin

• Sample space: $\Omega = \{\text{heads}, \text{tails}\}$
• Sample points: $\omega_1 = \text{heads}$, $\omega_2 = \text{tails}$

Example 2: the random experiment of rolling a die

• Sample space: $\Omega = \{1, 2, 3, 4, 5, 6\}$
• Sample points: $\omega_i = i$, $i = 1, 2, \dots, 6$

Example 3: the random experiment of measuring the length of an object

• Sample space: $\Omega = [0, +\infty)$
• Sample points: $\omega \in [0, +\infty)$

Random Events

Definition of a Random Event

Classification of Events

Classification:

1. Certain event: the sample space $\Omega$ itself
2. Impossible event: the empty set $\varnothing$
3. Elementary event: an event containing exactly one sample point
4. Compound event: an event containing several sample points

Representations of an Event

Representations:

1. Set representation: $A = \{\omega_1, \omega_2, \dots, \omega_n\}$
2. Descriptive representation: $A = \{\text{heads appears}\}$
3. Conditional representation: $A = \{\omega : \omega \text{ satisfies some condition}\}$

Examples

Example 4: rolling a die and defining events

• $A = \{\text{an even number appears}\} = \{2, 4, 6\}$
• $B = \{\text{a number greater than 3 appears}\} = \{4, 5, 6\}$
• $C = \{\text{a 1 appears}\} = \{1\}$ (an elementary event)

Relations and Operations of Events

Containment of Events

Properties:

• $A \subset A$ (reflexivity)
• If $A \subset B$ and $B \subset A$, then $A = B$ (antisymmetry)
• If $A \subset B$ and $B \subset C$, then $A \subset C$ (transitivity)

Equality of Events

Union of Events

Properties:

• $A \cup A = A$ (idempotence)
• $A \cup B = B \cup A$ (commutativity)
• $(A \cup B) \cup C = A \cup (B \cup C)$ (associativity)

Intersection of Events

Properties:

• $A \cap A = A$ (idempotence)
• $A \cap B = B \cap A$ (commutativity)
• $(A \cap B) \cap C = A \cap (B \cap C)$ (associativity)

Difference of Events

Complement Event

Properties:

• $A \cap \overline{A} = \varnothing$
• $A \cup \overline{A} = \Omega$
• $\overline{\overline{A}} = A$

Mutually Exclusive Events

Operation Laws of Events

Distributive laws:

• $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
• $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

De Morgan’s laws:

• $\overline{A \cup B} = \overline{A} \cap \overline{B}$
• $\overline{A \cap B} = \overline{A} \cup \overline{B}$

Examples

Example 5: the die-rolling experiment

Let $\Omega = \{1, 2, 3, 4, 5, 6\}$, and define the events:

• $A = \{\text{an even number appears}\} = \{2, 4, 6\}$
• $B = \{\text{a number greater than 3 appears}\} = \{4, 5, 6\}$
• $C = \{\text{a 1 appears}\} = \{1\}$

Computation:

• $A \cup B = \{2, 4, 5, 6\}$
• $A \cap B = \{4, 6\}$
• $A - B = \{2\}$
• $\overline{A} = \{1, 3, 5\}$
• $A$ and $C$ are mutually exclusive

Example 6: the coin-tossing experiment

Let $\Omega = \{\text{heads}, \text{tails}\}$, and define the events:

• $A = \{\text{heads appears}\} = \{\text{heads}\}$
• $B = \{\text{tails appears}\} = \{\text{tails}\}$

Properties:

• $A$ and $B$ are mutually exclusive
• $A \cup B = \Omega$
• $A = \overline{B}$, $B = \overline{A}$

Exercises

Exercise 1

Roll a die and define the events $A = \{\text{an even number appears}\}$, $B = \{\text{a number greater than 3 appears}\}$. Find $A \cup B$ and $A \cap B$.

Exercise 2

Let $\Omega = \{1, 2, 3, 4, 5\}$, $A = \{1, 2, 3\}$, $B = \{2, 3, 4\}$. Find $\overline{A}$ and $A - B$.

Exercise 3

Prove De Morgan’s law: $\overline{A \cup B} = \overline{A} \cap \overline{B}$.

Exercise 4

Let $A$ and $B$ be arbitrary events. Prove that: $(A \cup B) \cap \overline{A} = B \cap \overline{A}$.

Exercise 5

Let $\Omega = \{a, b, c, d\}$, $A = \{a, b\}$, $B = \{b, c\}$, $C = \{c, d\}$. Find $(A \cup B) \cap C$.