Probability

Context: FIT1058_MOC · a measure in $[0,1]$ of how likely an event is · outcomes get probabilities summing to $1$ · uniform finite case turns probability into counting

Quick Revision

• 🎯 Objective: $\mathrm{Pr}(A)\in [0,1]$ measures likelihood ➔ $\mathrm{Pr}(A)={\sum }_{x\in A}\mathrm{Pr}(x)$; uniform finite ⟹ $|A|/|U|$.
• 📦 Core Components: axioms (normalised weights) ➔ event = sum over outcomes ➔ Laplace ratio when equally likely.
• ⚡ Key Constraint: $|A|/|U|$ requires finiteness and uniformity; no uniform distribution on $\mathbb{N}$ exists.

📝 Core

1. The Measure (Axioms)

• Definition ➔ $\mathrm{Pr}(A)\in [0,1]$; impossible $=0$, certain $=1$.
• Axioms ➔ $0\le \mathrm{Pr}(x)\le 1$, ${\sum }_{x\in U}\mathrm{Pr}(x)=1$.
• Event probability ➔ $\mathrm{Pr}(A)={\sum }_{x\in A}\mathrm{Pr}(x)$ over the event’s outcomes.
• Graded vs binary ➔ a proposition is true/false; probability grades likelihood continuously.

2. Equally Likely Outcomes (Laplace / Uniform Case)

• Uniform formula ➔ finite $U$, all outcomes equal ⟹ each $\mathrm{Pr}(x)=\frac{1}{|U|}$ and $\mathrm{Pr}(A)=\frac{|A|}{|U|}$ — favourable ÷ total.
• Probability = counting ➔ reduces to finding $|A|$ and $|U|$; all counting techniques and inclusion–exclusion apply (divide the set formula by $|U|$).
• Worked micro-example ➔ fair die: $\mathrm{Pr}(\text{square})=\mathrm{Pr}(\{1,4\})=\frac{2}{6}=\frac{1}{3}$; $\mathrm{Pr}(\text{not }1)=\frac{5}{6}$.
• Refinement trick ➔ coarse non-uniform spaces (dice totals) hide a uniform substrate — drop to the $36$ equally-likely pairs first.

3. Non-Uniform & Infinite Spaces

• Unequal weights ➔ use the general sum: Scrabble $\mathrm{Pr}(\text{vowel})=0.42$ (sum of tile weights).
• Infinite $U$ ➔ needs a convergent sum: $\mathrm{Pr}(n)=\frac{1}{2^n}$ on $\mathbb{N}$ sums to $1$ (Geometric Series).
• No uniform on $\mathbb{N}$ ➔ constant $c>0$ sums to $\infty$, $c=0$ sums to $0$ — uniformity over an infinite set is impossible.

⚠️ Common Mistakes

• 💡 Only if equally likely ➔ dice totals are non-uniform, so $|A|/|U|$ is wrong on the 11 totals — refine to the uniform 36-pair space.
• 💡 No uniform distribution on $\mathbb{N}$ ➔ any constant weight fails normalisation; state finiteness before using Laplace.

🧠 Active Recall

State the axioms of a probability assignment and the equally-likely special case.

• Hint: Normalise, then sum.

Answer

• Short answer: $0\le \mathrm{Pr}(x)\le 1$, $\sum \mathrm{Pr}(x)=1$, $\mathrm{Pr}(A)={\sum }_{x\in A}\mathrm{Pr}(x)$; uniform finite ⟹ $|A|/|U|$.
• Why: Counting ratio ➔ uniform probability is favourable ÷ total (Laplace).

Show $\mathrm{Pr}(n)=1/2^n$ on $\mathbb{N}$ is valid, and why no uniform distribution on $\mathbb{N}$ exists.

• Hint: Normalisation.

Answer

• Short answer: ${\sum }_{n\ge 1}\frac{1}{2^n}=1$ (geometric); a constant $c$ gives $\infty$ or $0$.
• Why: Fails normalisation ➔ positive integers cannot be made equally likely.

Two fair dice: why is $\mathrm{Pr}(\text{total}=7)$ not $\frac{1}{11}$, and what is it?

• Hint: Refine to the uniform substrate.

Answer

• Short answer: Totals are non-uniform; on the 36 equally-likely ordered pairs, six give 7 ⟹ $\mathrm{Pr}=\frac{6}{36}=\frac{1}{6}$.
• Why: Uniformity check ➔ Laplace applies only where outcomes are interchangeable — pairs, not totals.