Sample Space and Events

Context: FIT1058_MOC · the set of all outcomes of a random experiment · events are its subsets · the universal set for Probability

Quick Revision

• 🎯 Objective: sample space $U$ = all outcomes; event = subset $A\subseteq U$ ➔ occurs when the outcome is in $A$.
• 📦 Core Components: experiment ➔ outcome ➔ $U$ (universal set) ➔ events as subsets.
• ⚡ Key Constraint: choice of space matters — a fine uniform space beats a coarse non-uniform one.

📝 Core

1. The Setup

• Experiment ➔ a partly random process yielding one outcome.
• Sample space ➔ $U$ = set of all outcomes = the universal set.
• Event ➔ subset $A\subseteq U$, occurs when the actual outcome lies in $A$.

2. Choosing the Space

• Non-uniform valid ➔ two-dice totals $\{2,\dots ,12\}$ (not equally likely).
• Uniform better ➔ $36$ ordered pairs, each $\frac{1}{36}$ (Cartesian Product).
• Fine expresses more ➔ “doubles” / “first die = 3” need the pair space.

3. Events Are Sets

• Operations ➔ $A\cup B$ (“or”), $A\cap B$ (“and”), $\overline{A}$ (“not”).
• Extremes ➔ $U$ = certain event, $\varnothing$ = impossible event.

Key identities:

$U=\{1,\dots ,6\}\times \{1,\dots ,6\},|U|=36(\text{uniform})$ $\text{vs totals }\{2,\dots ,12\}(\text{valid but non-uniform})$

⚖️ Core Decision Matrix

Space	Uniform?	Expressive?
$36$ pairs	yes ($\frac{1}{36}$)	high
$11$ totals	no	low
$U$	—	certain event
$\varnothing$	—	impossible event

When It Flips: the sample space is the universal set, which makes "not $A$" ($\overline{A}$) well-defined; a uniform fine space makes $|A|/|U|$ (Equally Likely Outcomes) usable, but accuracy of the model comes first.

📊 Exam Execution Trace

Applied Exercise

Problem: Give $U$, $A=$“total 8”, $B=$“doubles” for two fair dice. Derivation Proof / Hand-Calculation Walkthrough:

$$\begin{array}{rl}U& =\{1,\dots ,6{\}}^2,|U|=36\\ |A|& =5,|B|=6\end{array}$$

Final Extracted Output: $U$ = 36 pairs; $A$ has 5 outcomes, $B$ has 6 — both only in the fine space.

⚠️ Common Mistakes

• 💡 Coarse spaces lose information ➔ the $11$-total space can’t represent “doubles”; the fine $36$-pair space is both uniform and more expressive.

🧠 Active Recall

What are a sample space and event, and why prefer the 36-pair over the 11-total space?

• Hint: Uniform + expressive.

Answer

• Short answer: $U$ = all outcomes (universal set); event = subset; the pair space is uniform and can express doubles.
• Why: Finer = more ➔ totals are non-uniform and lose events like “first die = 3”.

Why can set operations be applied to events, and what are the certain/impossible events?

• Hint: Events are subsets.

Answer

• Short answer: $\cup ,\cap ,\overline{\cdot }$ build new events; $U$ certain (prob 1), $\varnothing$ impossible (prob 0).
• Why: Set footing ➔ lets probability reuse union/intersection and inclusion–exclusion.