Lecture 3 – Probability, Discrete RVs & Binomial Distribution (MATH1041)

Course & Chapter Context

• Course: MATH1041 – Statistics for Life and Social Sciences (Term 2, 2025)

• Chapter 3 focus: “Probability, Discrete Random Variables & the Binomial Distribution”

• Four lecture blocks covered in the transcript:

  • Lecture 1 – Probability

  • Lecture 2 – Discrete Random Variables

  • Lecture 3 – Means & Variances for Discrete Random Variables

  • Lecture 4 – The Binomial Distribution & Other Probability Models

Quick Regression Recap (link-back to previous week)

• Least-squares regression line: used to predict $\hat y$ from $x$; evaluate with residual plots.

• $r$ = correlation coefficient (strength & direction); $r^{2}$ = % of variation in $y$ explained by $x$.

  Example: $r=-0.78\,; r^{2}=0.61$ ⇒ 61 % of $y$’s variability explained by $x$.

• Residual $=y-\hat y$; want random scatter (homoscedasticity). Heteroscedasticity (“trumpet” shape) violates assumptions.

Why Study Probability?

• Random assignment & random sampling ⇒ data viewed as outcomes of random phenomena.

• Probability rules are the language for interpreting data produced by chance (forms basis for CI’s, hypothesis tests, ML, AI, etc.).

Fundamental Concepts & Vocabulary

• Random Phenomenon / Experiment: individual outcome unpredictable, but long-run pattern exists.

• Outcome ($\omega$): single possible result.

• Sample Space (S): set of all theoretical outcomes (e.g. $S={H,T}$ for one coin).

• Event (A): any subset of $S$. Occurs if outcome $\;\in\;$ A.

• Probability of outcome/event: long-run relative frequency.

Random vs Deterministic vs Haphazard

• Deterministic: outcome known in advance (e.g. sunrise).

• Random: unpredictable trial-by-trial, but regular long-run distribution.

• Haphazard: too irregular to model (no stable pattern).

Assigning Probabilities

1. Long-run empirical frequency (repeat experiment many times).

2. Physical symmetry / equally-likely outcomes ($p_i=1/L$).

3. Subjective / expert judgement.

Five Core Probability Rules

1. Boundedness: $0\le P(A)\le1$.

2. Covering Rule: $P(S)=1\;\;(\sum p_i=1)$.

3. Additive (General): $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$.

   • If A,B disjoint ⇒ $P(A\text{ or }B)=P(A)+P(B)$.

4. Complement: $P(A^c)=1-P(A)$.

5. Multiplication (Independence): A $\;\perp\;$ B iff $P(A\cap B)=P(A)P(B)\;\;(\Leftrightarrow P(B|A)=P(B))$ .

Conditional Probability & Independence

• Definition: $P(B|A)=\dfrac{P(A\cap B)}{P(A)}$ (requires $P(A)>0$).

• Think “A becomes the new sample space”.

• Tree-diagram helps: $P(A\cap B)=P(A)\,P(B|A)$.

• Independent $\;\ne\;$ Mutually Exclusive. Exclusive events cannot co-occur (intersection = 0), whereas independent events can but probability factorises.

Example (Student Enrolments):

• Table: 87 % full-time, 15 % international, 14 % both.

• $P(FT|Int)=\tfrac{0.14}{0.15}=0.93$ vs $P(FT)=0.87$ ⇒ not equal ⇒ dependence.

Random Variables (RV)

• Formal: function $X:S\to\mathbb R$ assigning a numerical value to each outcome.

• Upper-case letter = RV; lower-case = realisation (observed value).

• Discrete: countable list $x<em>1,x</em>2,\dots$ (often counts).

• Continuous: takes any value in an interval (measurements).

Probability Distribution of Discrete RV

• Table/pmf $p<em>X(x</em>k)=P(X=x<em>k)$ satisfying $\sum p</em>X(x<em>k)=1$ & $p</em>X(x_k)\ge0$.

• Graph: probability mass plot (vertical spikes).

Examples

• Spinner game payouts: $x\in{-4,2,6},\;p=(3/8,4/8,1/8)$ .

• Two dice sum $Y$: pmf with 11 possible sums (2–12).

Key Specialized Discrete Distributions

• Uniform (discrete): $P(X=r)=1/L$ for $r=1,\dots,L$ .

• Binomial $B(n,p)$.

• Poisson $Pois(\lambda)$ (rare events / rate).

• Hypergeometric $HGeom(m,N-m,n)$ (sampling without replacement).

• Geometric (failures before 1st success with prob. p).

Mean (Expectation) & Variance of Discrete RV

• Mean (expected value): $\mu<em>X=E[X]=\sum</em>{k}x<em>k p</em>k$ .

• Variance: $\sigma<em>X^{2}=E[(X-\mu</em>X)^{2}]=\sum<em>{k}(x</em>k-\mu<em>X)^{2}p</em>k$ .

• Std-dev $\sigma<em>X=\sqrt{\sigma</em>X^{2}}$ .

Linear Combination Rules (always hold)

• For constants a,b: $E[a+bX]=a+b\mu<em>X$ . • For two RVs: $E[X\pm Y]=\mu</em>X\pm\mu_Y$ .

Variance Rules (need independence except scaling)

• Scaling: $Var(a+bX)=b^{2}Var(X)$.

• Add/Sub independent: $Var(X\pm Y)=Var(X)+Var(Y)$ (note “+” in both cases).

• Std-dev does not add linearly.

Law of Large Numbers (LLN)

• For i.i.d. $X<em>1,\dots,X</em>n$ with mean $\mu$: sample mean $\bar X = \frac{1}{n}\sum X_i$ converges to $\mu$ as $n\to\infty$ .

• Justifies interpreting probability & expectation as long-run averages.

The Binomial Distribution $B(n,p)$

Conditions (Bernoulli trials):

1. Fixed number n of trials.

2. Each trial results in Success/Failure.

3. Trials independent.

4. Constant success probability p.

Probability mass function:

$\quad P(X=x)=\binom{n}{x}p^{x}(1-p)^{n-x},\quad x=0,1,\dots,n.$

Key moments:

$\quad E[X]=np,\qquad Var(X)=np(1-p),\qquad \sigma=\sqrt{np(1-p)}.$

Counting Tool $\binom{n}{r}$

$\binom{n}{r}=\frac{n!}{r!(n-r)!}$ = number of unordered subsets of size r.

Practical Examples

• True/False test (n=5, p=0.5): $P(\text{score}\ge3)=0.5$ .

• Inspector weighing 10 cereal boxes (p=0.2 defective): $P(X\ge2)=1-[P(0)+P(1)]\approx0.624$ (used complement rule).

Other Discrete Models (Quick Reference)

• Poisson($\lambda$): count of rare independent events in fixed interval.

  • pmf $P(X=k)=e^{-\lambda}\,\lambda^{k}/k!$; mean = var = $\lambda$.

• Hypergeometric: without-replacement sampling; parameters (white m, black b, draws n).

  • $P(X=x)=\frac{\binom{m}{x}\binom{b}{n-x}}{\binom{m+b}{n}}$.

• Geometric(p): # failures before 1st success.

  • pmf $P(X=x)=(1-p)^{x}p$; $E[X]=\frac{1-p}{p},\;Var(X)=\frac{1-p}{p^{2}}$.

Worked-Example Gallery

1. Left-handed lunch group (n=3, p=0.12):

   • $P(X=2)=3\times0.12^{2}\times0.88\approx0.038$.

   • P(X\le2)=1-P(3)=1-0.12^{3}\approx0.998$.

2. Social-media dual users (n=5, p=0.6):

   • E[X]=3,\;\sigma^{2}=1.2,\;\sigma\approx1.10$.</p></li><li><p><strong>Seed-weighing with measurement error$\sigma=10\,\text{mg}$</strong>: averaging two weighs halves variance ⇒$\sigma_{\bar X}=7.07\,\text{mg}$.</p></li><li><p><strong>Double die roll</strong>:</p><p>• Single die$Var=2.916$; sum of two independent ⇒$Var=5.832$.</p></li></ol><div data-type="horizontalRule"><hr></div><h5 id="1bebab39-7c5e-41df-9eaa-b146dc030e59" data-toc-id="1bebab39-7c5e-41df-9eaa-b146dc030e59" collapsed="false" seolevelmigrated="true">Modelling Checklist</h5><ol><li><p>Identify outcome, sample space, event(s).</p></li><li><p>Decide if equally-likely <em>or</em> assign probabilities empirically/symmetry.</p></li><li><p>Translate English to probability notation.</p></li><li><p>Test binomial conditions; if met, store$n,p.

3. Compute desires via pmf / complement / cumulative functions.

4. Interpret in context (units, risk, practical meaning).

Common Pitfalls & Ethical Notes

• Multiplying probabilities requires independence (Sally Clark SIDS case – tragic mis-use).

• “Mean” could denote population mean (\mu$), sample mean ($\bar X$RV), or observed mean (number). Clarify!</p></li><li><p>Risk$\;\ne\; expected profit; variance matters in decisions.

• Always plot data before fitting models (Anscombe’s, Greta’s regression caution).

Study & Exam Tips

• Re-derive formulas once without notes – embeds memory.

• Practise tree diagrams and calculator/R functions (dbinom, pbinom, choose, rnorm, etc.).

• Build small simulations to verify analytical answers (R replicate loops).

• Prepare a personal glossary (keywords slides pp 219-220).

• Use “Seeing Theory” interactive site for visual intuition.

Key Equations (Quick Sheet)

• P(A\cup B)=P(A)+P(B)-P(A\cap B)$</p></li><li><p>$P(A^c)=1-P(A)$</p></li><li><p>$P(B|A)=\frac{P(A\cap B)}{P(A)}$</p></li><li><p>$E[X]=\sum xk pk$</p></li><li><p>$Var(X)=E[(X-\mu)^2]$</p></li><li><p>$\binom{n}{r}=\frac{n!}{r!(n-r)!}$</p></li><li><p>$P_{\text{Bin}}(X=x)=\binom{n}{x}p^{x}(1-p)^{n-x}$</p></li><li><p>$E{\text{Bin}}=np,\;Var{\text{Bin}}=np(1-p)$</p></li><li><p>$P_{\text{Pois}}(X=k)=e^{-\lambda}\,\lambda^{k}/k!$$

Reflective Questions

• Can I explain (without symbols) the difference between independent & mutually exclusive?

• Given a problem, what clues signal a binomial vs hypergeometric vs Poisson model?

• How does the Law of Large Numbers justify quality-control sampling?

• In what way do variance rules change when RVs are dependent?

End of consolidated notes.