Maths Boot Camp: Session 4

Application 1 — Riker (1962)

• Counting possible coalitions in a legislature
• Size principle: coalitions form at the smallest size needed
• Minimum winning coalition

Block 1 — Key Formulas at a Glance

Four building blocks, all we need for the rest of this block:

Block 1 — Bringing It Together

A committee-selection example, combining both tools from this block:

• A legislative committee needs 3 members chosen from 8 eligible MPs, then those 3 must be ranked as first, second, and third speaker in a debate
• Step 1 — Combinations: how many distinct groups of 3 MPs can be chosen?

• Step 2 — Permutations: each chosen group of 3 can be ranked in 3! = 6 ways

The Paradox of Voting

• Rational-choice logic: vote only if the expected utility of voting is positive

U = (P \times B) - C

• P = probability your vote is decisive; B = benefit if your candidate wins; C = cost of voting
• Utility of voting rests on the probability that your vote is the one that decides the outcome
• In large elections, P \approx \frac{1}{n} — probability of being decisive diminishes rapidly as the electorate n grows
• Larger pool of voters → smaller individual probability of being pivotal → individual participation looks non-rational
• Yet people vote anyway: the paradox

Definition of Probability

Probability measures the likelihood of an event, a number between 0 and 1:

• 0 means the event is impossible; 1 means the event is certain

For an event A in a sample space S: 0 \leq P(A) \leq 1. The sample space has total probability 1 (the sum of all probabilities).

Example: for a fair coin toss, P(\text{Heads}) = \frac{1}{2}

• Rolling a fair die: 6 equally likely outcomes, each with probability \frac{1}{6}, summing to 1

Axioms of Probability

• Three fundamental axioms:
  • Non-Negativity: P(A) \geq 0
  • Normalisation: P(S) = 1
  • Countable Additivity: for mutually exclusive E_1, E_2, \dots

P\left(\bigcup_{i=1}^{\infty} E_i\right) = \sum_{i=1}^{\infty} P(E_i)

Reading this formula: the “\dots” just means “as many mutually exclusive events as you like” — could be 3, could be infinitely many. \bigcup is just union — “E_1 or E_2 or E_3 or …” — stacked across however many events there are, and \sum adds up their probabilities in that same order. With only 3 events, this reads simply as P(E_1\cup E_2\cup E_3) = P(E_1)+P(E_2)+P(E_3).

Joint and Conditional Probability

• Two ways of combining two events’ probabilities — building blocks for Bayes’ Theorem next
• Joint: both happen, P(A \cap B)
  • Independent: P(A \cap B) = P(A)\cdot P(B)
  • Mutually exclusive: P(A \cap B) = 0
  • General union: P(A \cup B) = P(A)+P(B)-P(A \cap B)
• Conditional: probability of A given B has occurred

P(A|B) = \frac{P(A \cap B)}{P(B)}, \qquad P(B) > 0

• If independent, conditioning tells you nothing new: P(A|B) = P(A)

Law of Total Probability

Calculates P(A) using a partition \{B_1,\dots,B_n\} of the sample space — mutually exclusive (B_i\cap B_j=\emptyset) and exhaustive (\bigcup_i B_i = S):

P(A) = \sum_{i=1}^n P(A|B_i)\,P(B_i)

Example: a campaign reaches voters via 3 methods — canvassing (B_1), mail (B_2), digital ads (B_3) — a partition of all contacted voters. Let A = “the voter turns out to vote.”

• P(B_1)=0.4, P(B_2)=0.3, P(B_3)=0.3 (share reached by each method);   P(A|B_1)=0.9, P(A|B_2)=0.7, P(A|B_3)=0.8 (turnout given each method)

Bayes’ Theorem — Derivation

Start from the definition of conditional probability, two ways:

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad \Rightarrow \quad P(A \cap B) = P(A|B)\cdot P(B)

P(B|A) = \frac{P(A \cap B)}{P(A)} \quad \Rightarrow \quad P(A \cap B) = P(B|A)\cdot P(A)

Since both expressions equal P(A \cap B), set them equal and solve for P(A|B):

P(A|B)\cdot P(B) = P(B|A)\cdot P(A) \quad \Rightarrow \quad P(A|B) = \frac{P(B|A)\cdot P(A)}{P(B)}

Bayes’ Theorem — Worked Example

Before any evidence, you hold a prior belief — new evidence updates it.

Example 1 — Medical Test: P(D)=0.01 (1% of people have the disease), P(+|D)=0.99 (true positive rate), P(+|\text{no }D)=0.05 (false positive rate)

Bayes’ Theorem — Worked Example

Let’s run the same recipe on a very different kind of evidence — a political exit poll.

Example 2 — Exit Poll: you already believe Candidate A is more likely to win, from polling. An exit poll comes in — how should it change your belief?

Bayes’ Theorem — Worked Example

Continuing the exit poll:

Boole’s Inequality — Worked Example

Three bills in Congress: \Pr(A_1)=0.3, \Pr(A_2)=0.4, \Pr(A_3)=0.5 (probabilities each bill passes)

Block 2 — Bringing It Together

A content-moderation example, combining both tools from this block:

• 10% of viral posts are actually false; 90% are true
• A flagging algorithm catches 90% of false posts, but also flags 8% of true posts
• If a post gets flagged, what’s the probability it’s actually false?

Levels of Measurement

Data is classified into four levels:

• Nominal (Categorical): no meaningful order — e.g. political affiliation, gender
• Ordinal: meaningful order, no consistent scale — e.g. party ID left-right, Likert responses
• Interval: consistent differences, zero is arbitrary — e.g. dates, temperature (°C/°F)
• Ratio: interval properties plus a true, meaningful zero — e.g. crime rates

Probability Mass Function (PMF)

• In plain terms: a full list of every possible outcome, and how likely each one is
• For a discrete RV, the PMF gives the probability of each specific value: P(X=x) = p(x)
• Lets us read off any probability directly, and is what we’ll sum (next) to get the expectation
• Properties: 0 \leq p(x) \leq 1, and \sum p(x) = 1

Example: tossing a coin, X=0 for tails, X=1 for heads: P(X=0)=0.5, P(X=1)=0.5

Probability Density Function (PDF)

• In plain terms: like a PMF, but for outcomes that can take any value in a range, not just a fixed list
• For a continuous RV, the PDF gives density at a point, not a probability directly: f(x) \geq 0, \int_{-\infty}^{\infty} f(x)\,dx = 1
• Probability at any single point is zero (a point has no width)
• To get an actual probability, integrate the PDF over an interval

Expectation of a Discrete Random Variable

E(Y) = \sum_{y} y \cdot \Pr(Y=y) = \sum_y y \cdot p(y)

Example: expected value of a fair die roll:

Measures of Random Variables — Mean

• The mean (\mathbb{E}[Y]): the central tendency of a distribution
• Same idea, discrete or continuous:

\mathbb{E}[Y] = \sum_i y_i \cdot \Pr(Y=y_i) \quad \text{(discrete)}, \qquad \mathbb{E}[Y] = \int_{-\infty}^{\infty} y \cdot f_Y(y)\,dy \quad \text{(continuous)}

Variance and Standard Deviation

• The mean tells us the centre of a distribution — but says nothing about its spread
• Deviation from the mean: y - \mathbb{E}[Y] — squared, to stop positive and negative deviations cancelling out: (y-\mathbb{E}[Y])^2
• Variance: the average squared deviation — \text{Var}(Y) = \mathbb{E}\left[(Y-\mathbb{E}[Y])^2\right]
• Standard Deviation: \sigma_Y = \sqrt{\text{Var}(Y)} — same units as Y, unlike variance

Remember from the R Bootcamp

You’ve already worked through this — the exact same walkthrough from R Session 2, using real UK road-accident data.

Example Calculation — Variance and SD

Ideology scores (0–100, left–right) for three candidates: 50, 60, 70. Mean \mathbb{E}[Y]=60.

• For 50: (50-60)^2=100; for 60: (60-60)^2=0; for 70: (70-60)^2=100

Example Calculation — Population or Sample?

Same three ideology scores — 50, 60, 70 — but does it matter whether they’re the whole population or just a sample?

Example — Same Mean, Different Spread

A get-out-the-vote (GOTV) example, putting mean and variance to work together:

• Two GOTV messages are tested; observed turnout lift (%) across 3 similar precincts:
  • Message A: 2, 4, 6
  • Message B: 0, 4, 8

• Same average effect — but the underlying spread is very different

• Message B is riskier: same expected lift, much less predictable across precincts

Covariance and Correlation

• In plain terms: do two things tend to move together, and how strongly?
• Covariance: how two random variables vary together — \text{Cov}(X,Y) = \mathbb{E}[(X-\mathbb{E}(X))(Y-\mathbb{E}(Y))]
  • >0: tend to increase together; <0: move in opposite directions
• Correlation: covariance normalised by both SDs — \rho_{XY} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}
  • Always between -1 and +1; closer to either end means a stronger linear relationship

Example Calculation — Covariance and Correlation

Ad spending (X, $10k) and vote share (Y, %) for three candidates: (1,45), (2,55), (3,50). Means: \bar{X}=2, \bar{Y}=50.

• Deviations: X-\bar{X} = -1, 0, 1; \quad Y-\bar{Y} = -5, 5, 0

Normal Distribution

• Describes data clustering around a mean; mean, median, and mode are all equal
• Symmetric and bell-shaped
• The 68-95-99.7 Rule:
  • ~68% of data falls within one standard deviation of the mean
  • ~95% within two standard deviations
  • ~99.7% within three standard deviations

Quick Check — Random Variables (cont.)

Block 3 — Bringing It Together

A GOTV campaign wants to know: does spending more actually buy a bigger turnout lift? Spending ($1,000s) and observed lift (%) across 3 precincts: (1,3), (2,6), (3,6).

• A strong positive relationship: more spending tends to come with a bigger lift — but \text{Var}(\text{Lift})=2 says that lift is still not perfectly predictable from spending alone