for the general cheatsheet (Units 1-5: proof rules, sets, induction), see the TCX1004 cheatsheet. full unit-by-unit notes in the TCX1004 notebook.
Exam: Apr 30, 2026 · 17:00 · LT7A Seat 64 · Open Book · No calculators · 1.5h Scope: Unit 6+ (Combinatorics, Graph Theory, Probability, Distributions/Expectation/Variance) Note: leave answers unsimplified.
Unit 6: Combinatorics
Placeholder — to be translated from Unit Notes.
Unit 7: Graph Theory
Placeholder — to be translated from Unit Notes.
Unit 8: Probability
Placeholder — to be translated from Unit Notes.
Unit 9: Distributions, Expectation, Variance
Step 1 — identify the distribution (3 questions)
- Single trial, two outcomes? → Bernoulli$(p)$
- Fixed $n$ trials, count successes? → Binomial$(n, p)$ — verify BINS (Binary / Independent / Number fixed / Same $p$)
- Count trials until first success? → Geometric$(p)$
If none match: likely Uniform (all outcomes equally likely), or distribution is unknown → use bounds (Markov / Chebyshev).
Step 2 — distribution formulas
| Distribution | $E[X]$ | $\text{Var}[X]$ | Trigger words |
|---|---|---|---|
| Bernoulli$(p)$ | $p$ | $p(1-p)$ | one trial, two outcomes |
| Binomial$(n, p)$ | $np$ | $np(1-p)$ | $n$ fixed trials, count successes (BINS) |
| Geometric$(p)$ | $1/p$ | $(1-p)/p^2$ | until first success |
| Uniform (discrete) | $(a+b)/2$ | depends on range | equally likely outcomes |
Bounds (distribution unknown)
| Bound | Formula | When to use |
|---|---|---|
| Markov | $P[X \geq a] \leq E[X]/a$ | only $E[X]$ known, one-tail, $X \geq 0$ |
| Chebyshev | $P[\lvert X - \mu \rvert \geq a] \leq \text{Var}[X]/a^2$ | $E[X]$ and $\text{Var}[X]$ known, two-tail |
Linearity of Expectation
$$E[X + Y] = E[X] + E[Y] \quad \text{(always, even if dependent)}$$
$$\text{Var}[X + Y] = \text{Var}[X] + \text{Var}[Y] \quad \text{(only if independent)}$$
Variable legend (don’t confuse under pressure)
| Symbol | Source | Role |
|---|---|---|
| $\mu$ | $E[X]$ from question | center |
| $\text{Var}[X]$ | from question | spread$^2$ |
| $\sigma$ | $= \sqrt{\text{Var}[X]}$ (derived) | std dev — appears in $1/k^2$ form only |
| $a$ | threshold inside $P[\lvert X - \mu \rvert \geq a]$ | denominator (squared) |
| $k$ | $= a/\sigma$ | $\sigma$-multiples in alternate form |
$a$ and $\sigma$ are different variables even when they share a numeric value.
Pre-submit sanity checks
- probability $\in [0, 1]$ — never negative
- $E[X] \in [\min X, \max X]$ — NOT bounded by $[0, 1]$
- re-add arithmetic on 1-mark questions
- plugged $a$ (threshold) or $\sigma$ (std dev)? they are different
- Chebyshev monotonicity: larger $a$ → smaller bound. if bound grew, you plugged wrong.