Sample Spaces and Events

Experiments and Outcomes

Probability theory provides mathematical tools for reasoning about uncertain events. The starting point is the notion of an experiment: a process that produces an outcome.

Examples:

• Rolling a die: outcome is which face lands uppermost
• Flipping a coin three times: outcome is a sequence like HTH
• Measuring the time until a component fails: outcome is a non-negative real number

Sample Space

The sample space $\Omega$ (or $S$) is the set of all possible outcomes of an experiment.

Discrete Sample Spaces

A sample space is discrete if it is finite or countably infinite.

Example (coin flip): $\Omega = \{H, T\}$

Example (two dice): $\Omega = \{(i,j) : i,j \in \{1,2,3,4,5,6\}\}$ has 36 outcomes.

Example (coin flips until heads): $\Omega = \{1, 2, 3, \ldots\}$ is countably infinite.

Continuous Sample Spaces

A sample space is continuous if outcomes can take any value in an interval.

Example (time to failure): $\Omega = [0, \infty)$

Example (temperature): $\Omega = \mathbb{R}$ or some bounded interval.

Events

An event is a subset of the sample space: a collection of outcomes. We say an event $E$ occurs if the actual outcome of the experiment is in $E$.

Example (single die):

• Event “rolling an even number”: $E = \{2, 4, 6\}$
• Event “rolling at least 5”: $F = \{5, 6\}$

Example (two dice):

• Event “sum equals 7”: $E = \{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}$
• Event “both dice show the same”: $F = \{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\}$

Special Events

• The entire sample space $\Omega$ is the certain event: it always occurs.
• The empty set $\emptyset$ is the impossible event: it never occurs.
• An elementary event (or atomic event) contains exactly one outcome: $\{x\}$ for some $x \in \Omega$.

Event Operations

Since events are sets, we can combine them using set operations:

Operation	Notation	Meaning
Union	$E \cup F$	$E$ or $F$ (or both) occurs
Intersection	$E \cap F$ or $EF$	Both $E$ and $F$ occur
Complement	$E^c$ or $\bar{E}$	$E$ does not occur
Difference	$E - F$ or $E \setminus F$	$E$ occurs but $F$ does not

Mutually Exclusive Events

Events $E$ and $F$ are mutually exclusive (or disjoint) if they cannot both occur: $E \cap F = \emptyset$.

Example (single die): The events “roll an even number” $\{2,4,6\}$ and “roll a 3” $\{3\}$ are mutually exclusive. But “roll an even number” and “roll at most 4” $\{1,2,3,4\}$ are not mutually exclusive: both occur if we roll 2 or 4.

The Event Space

The collection of all events forms a sigma-algebra (or $\sigma$-algebra), denoted $\mathcal{F}$. This is a collection of subsets of $\Omega$ satisfying:

1. $\Omega \in \mathcal{F}$
2. If $E \in \mathcal{F}$, then $E^c \in \mathcal{F}$ (closed under complements)
3. If $E_1, E_2, \ldots \in \mathcal{F}$, then $\bigcup_{i=1}^{\infty} E_i \in \mathcal{F}$ (closed under countable unions)

For finite sample spaces, $\mathcal{F}$ is typically the power set $\mathcal{P}(\Omega)$: all possible subsets.

Why sigma-algebras? They ensure we can assign probabilities consistently. If we can talk about the probability of $E$ and of $F$, we should also be able to talk about the probability of $E^c$ and $E \cup F$.

De Morgan’s Laws

For any events $E$ and $F$:

$(E \cup F)^c = E^c \cap F^c$ $(E \cap F)^c = E^c \cup F^c$

Intuition: For $E \cup F$ to not occur, both $E$ and $F$ must not occur. For $E \cap F$ to not occur, either $E$ or $F$ must not occur.

Partition

A partition of $\Omega$ is a collection of mutually exclusive events $F_1, F_2, \ldots, F_n$ whose union is $\Omega$:

$F_i \cap F_j = \emptyset \text{ for } i \neq j$ $\bigcup_{i=1}^{n} F_i = \Omega$

Partitions are crucial for the Law of Total Probability.

Example (single die): $\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}$ partitions the sample space.

Example (blood type): $F_1 = \text{type A}, F_2 = \text{type B}, F_3 = \text{type AB}, F_4 = \text{type O}$ partitions by blood type.

Summary

Concept	Definition
Sample space $\Omega$	Set of all possible outcomes
Event $E$	Subset of the sample space
Elementary event	Event containing one outcome
Mutually exclusive	$E \cap F = \emptyset$
Partition	Mutually exclusive events that cover $\Omega$